Презентация «Инварианты конечных групп»

Дипломная работа

Инварианты конечных групп

Выполнила учитель математики и физики гимназии № 42

Филиппова Мария Сергеевна

Теория инвариантов групп изучает алгебраические выражения, изменяющиеся определённым образом при невырожденных линейных заменах переменных. Множество iVBORw0KGgoAAAANSUhEUgAAAA0AAAAZCAYAAADqrKTxAAAACXBIWXMAAA7OAAAOxgEGhk// AAABHElEQVR4nO2TvW3DMBCFHwEPQiGNCw9BwQ0Nt9mAbIK0EjKDIbVBGnoD91JjWEOoSBOI m1xIMfo3HLsMkFcK9929dzytiAiPavUw8VehKiUW57OPEqYpoDjYDLJ03EXQpS+gtiD0YBTn GzzxK5OqtAOKHvAS+wSo1+ALe86SdyTNxwQIVMZILDI5WwefIcGb4vhNAbJnnErP7OEasptE B9nzCYERt6sn0KdHJNYTZxWlLEa/+OQCygQb7LUaVhokWNYYqiONUho02eDirouQz1u/7j7r CKrxZV3/0bSQ1b3bdmIhQOLVQOYa+r2C6mzYI73odj0z290krljRgHZRDJYj/JU+xyVBdBgu YWnPg6QWecaXcCXT/fqHfvQNY+dgcNquFeAAAAAASUVORK5CYII= всех невырожденных линейных преобразований переменных, реализуемых в виде невырожденных квадратных матриц, образует группу, которая является подгруппой полной линейной группы.

Многочлен iVBORw0KGgoAAAANSUhEUgAAAGEAAAAZCAYAAAAhd0APAAAACXBIWXMAAA7EAAAOxgGnHWyZ AAAEJklEQVR4nO1ZMY7qMBB9ljiIEQ3FHiKIJoh2b5A0iDaIM6xIu6IJN6BFSYPIISjSIHKT +WObBBKS3ewnCFbLq9YhGc/4jWeevR0iwguPRefRDrzwIuEp8CLhCdAiCTHNxAD74IjQkaI9 u78f6WpEXfcNO1rAAq7WphUS4pmgge/xJIRFxSR/HdIJBTkqSQU+giOVk/RmEhTLhoBqll/I YIkF7ZiILkYoEnEbCemKJm4Ebxe+CGgEJmLnkRhMsBqG5EizZjeREH+6iOwAS6sVD/8GrDE8 +FhvUzjMgkKJhJRWoy6SOdd2i1mKZ8yaf/rNRnAMkbGnGvFG/eT1IGt2gekV6tMAx9Ax72U2 vR1oYT397mk/BkuMPZC/3iJ1HFL2ziRwaRl1XUCpm5yAPQ+JFz6FIsedrDAMzYdID9gr3/qy 1vnNmEDjGcRgjW3qwAHPsRmDaPH0i69wrxhk3wb8kz2Z7wTu3BkBumHwmJm2eWwyX6L3BoKf IFVG1CdpgghqI8jKiawFCUv/NSa1/ZKUifxIMOds+i24VwySFxM6hQ00CfFswFOwwjnVKMQb HnP5Gcri13YfpSdNpoQi3h10effyrvqVDfweMURMqjatSDC13Q6mucJJD4qlN5yT/Pv6/0UA Qu8iBJhat7v+GNwjBhtZJe9ktf2MlLZrLjTe/Cw79c7gR+MLD2SfzfCmOqS8b2X9XNxrPnRv vyhlFdANEE/arBvG0NjcobjiHS5QUBXKz/YGT5MwB/Z7NpXpD0odFLLg6rvK6Wg14Rp6DADu OZuYD3RW6Q19pI/MwPvfsO6J72P4sUW9wAGyat9Rkmka2OS7A4z6fJLrXfSDk2KKVL/I5FmO k9TaH5g2i2RFmYpnRu46MsaGx/4mVtKXT40bbnWnE/ZwyUpDCk1GUu30+Vrk+lSeSUhVq7Ws zny+kJQZ0XbFvdZXtpvFcFKOkZHwvU+BK0lbtHgq/8P8N92Y9d1GT0nSbv6F2xVwta0jqOZC zpoGsDngz9hRjuXIgs4Xhl3NiBalgKV8zsu+pjHEswlheUQwmSD5HCFRkna6wqhbU7oqRM/5 nGAtBGtfkxn7OhZLkI5YBmvqDmYqK8jKFlZfWJVfvX72I9QoMyUj2e2CT2Fpom/nrrDdNAZr EQpLlewowv49O2MlFLHN5dVEZelvcHViNsKouQpSju0SJo7f3l0Q0R7M9vV2DZLiUbZ1dp8l fsxGbSakaLP+qr9Mgm7K3tz6kQ86G6dci2uuaptANyvMr56rMwzyktAu2rKt1U6uJk0iv40l l7QVpQ4TvBrB/D+h+qq/SEJWr+R/eFJRBhoFcKmOoOot6LKBnk+t7aMd20bS2+/F4uNzadBZ rxbdCfHV0hRJ0H3hZq9+hJt7xcMhhRPqU3TN+Hu8/sf8BHiR8AT4B+W5heUasoIcAAAAAElF TkSuQmCC из кольца многочленов iVBORw0KGgoAAAANSUhEUgAAAFwAAAAZCAYAAAC8ekmHAAAACXBIWXMAAA7FAAAOxgFsQb88 AAADRElEQVR4nO1ZPW6zQBB9K/kgi9Kk8CFANFhufYOlsdIa+QyR3UZuzA3cItxY4RAuaD6Z m8w3u0CME5Jg/IOT+DUxKzQ7+2Z585MeEeGO66HXtQN/DXfCr4w74VdGC8IzCgcW/PV+ZfJK mNkQ53Pr5yMLB2RVSPKWO8RKil7dC40InLyCZvad5E8gVSxI6V8JBcLBtljvVV/YQZPex9Du wsW/gQNJyVK+4d4IsiNn/gIqhGf0b6v5djXhd6m4EPaEZxus+IL3p7J4Dmlg+ShUHa80g90g EEkgyJlDZwnsYpUHLwlI6MUfoPuX9r9CeMrkehjJ4lm6GHngtcrGDZyNhgQaBhDOCptMQYED Fw1BNLtpojWu4f8b4Uk0N1F1Zf6chWP4/eMias9I2ObXkCaYI80yhM8pprE6h68XxzX8LwjP 9Rv9B77JvMnAotVox2TLllGVeOSvw3csU2KqH5cTLud/STh0gYK1A6H1i6VluZAnmJXioQ8C y9GTfaqLXeBy/ueEJxE0z6bhkSF0slxtMigl21nlhPtsApdyKPFpmWkSFG4wkTb0vw0M4ZnR k0nR8OTJ0l9tkClF8ujPiVv/Mevebglw4KKEqxv73RvVtndy2gHOj+/9PwU9vcFmVW14pHBH HjHjJku7mwGNsTBzgCYGk8BCOmXdkwkifp5HiR4TcHsbcSoqSkt3wVlfCkN8WmdDl2b1pWhZ tr2NH8rytVLGlQEt5xdNbTfz3+Q4+GuW3V2MhxeBD2XkN4Qb/faW+4ZHuiNWcR++JWqdrkN5 yP0cxsbT0qO5r/PC4QGlbJuML4em/ifBmLDYYTkeI30ZINVl5JOW4Wbyw4TbYvb+vz5SiZiO K4X2w5qv1xrDe6x1XpdufPbqJh98/XbfGttN/bdnsbD1QGq9xpYruVgHJ0lpzTYXX52nwA3O wxOKjGQ0a7Y6sW2KDL71RVGhexiPyW9isz3hc1NCmk+j7TzcDMswPVhLAge40Hz9XLZNkTGZ FhKZ9zD9IXcwYUhcaJj1D/Pw4m8LwqVQsWkG2jt84EweuDJX7Lu98+M8tvMiwxsdCsjcEdjq M5R56hNZ60RSTtL2zvH+wh13AW9Qw3837oRfGf8BouvvHHIwLKoAAAAASUVORK5CYII= , где k – поле, не меняющийся при всех преобразованиях группы iVBORw0KGgoAAAANSUhEUgAAAA0AAAAZCAYAAADqrKTxAAAACXBIWXMAAA7OAAAOxgEGhk// AAABHElEQVR4nO2TvW3DMBCFHwEPQiGNCw9BwQ0Nt9mAbIK0EjKDIbVBGnoD91JjWEOoSBOI m1xIMfo3HLsMkFcK9929dzytiAiPavUw8VehKiUW57OPEqYpoDjYDLJ03EXQpS+gtiD0YBTn GzzxK5OqtAOKHvAS+wSo1+ALe86SdyTNxwQIVMZILDI5WwefIcGb4vhNAbJnnErP7OEasptE B9nzCYERt6sn0KdHJNYTZxWlLEa/+OQCygQb7LUaVhokWNYYqiONUho02eDirouQz1u/7j7r CKrxZV3/0bSQ1b3bdmIhQOLVQOYa+r2C6mzYI73odj0z290krljRgHZRDJYj/JU+xyVBdBgu YWnPg6QWecaXcCXT/fqHfvQNY+dgcNquFeAAAAAASUVORK5CYII= , называется инвариантом этой группы. Совокупность всех инвариантов группы iVBORw0KGgoAAAANSUhEUgAAAA0AAAAZCAYAAADqrKTxAAAACXBIWXMAAA7OAAAOxgEGhk// AAABHElEQVR4nO2TvW3DMBCFHwEPQiGNCw9BwQ0Nt9mAbIK0EjKDIbVBGnoD91JjWEOoSBOI m1xIMfo3HLsMkFcK9929dzytiAiPavUw8VehKiUW57OPEqYpoDjYDLJ03EXQpS+gtiD0YBTn GzzxK5OqtAOKHvAS+wSo1+ALe86SdyTNxwQIVMZILDI5WwefIcGb4vhNAbJnnErP7OEasptE B9nzCYERt6sn0KdHJNYTZxWlLEa/+OQCygQb7LUaVhokWNYYqiONUho02eDirouQz1u/7j7r CKrxZV3/0bSQ1b3bdmIhQOLVQOYa+r2C6mzYI73odj0z290krljRgHZRDJYj/JU+xyVBdBgu YWnPg6QWecaXcCXT/fqHfvQNY+dgcNquFeAAAAAASUVORK5CYII= образует подалгебру алгебры многочленов от iVBORw0KGgoAAAANSUhEUgAAAAsAAAAZCAYAAADnstS2AAAACXBIWXMAAA7EAAAOxgGnHWyZ AAAA8UlEQVR4nO2TOw6CQBRF70tYCMbGwkUMsdHQsgNYAoQ1kLE1NrIDWyONcRZhQUNgJ8+Z 4RP8FCQWWviqycmZO29+DjNjajmTzb/8sawSYm9rRjGuLCGgkJAHi9YH1OcQLkBOviEuUwb7 Cci7oVI5Mq9Eqm9WKsMi7FQIKXRyeGayayjoey8QZQFqnW6SNNVsjYX71HNT3cyaOOzDXuzY EvNHueHLsbD9rdxhescC9KiXUVp3NaS+Y62sTnbn8XyIfcuccb+LkdtWx1TCdPI7uXzuV5fw 9alvEc0IUXwFS0FWFpLp9b8IkuasR+SXHtL35TvY3F8yzJa25AAAAABJRU5ErkJggg== - переменных.

Классическая постановка основной задачи теории инвариантов - фактически вычислить инварианты данной группы, а также дать полное их описание.

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Рассмотрим отображение:

:.

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Определение: множество полиномов, инвариантных относительно группы iVBORw0KGgoAAAANSUhEUgAAAA0AAAAZCAYAAADqrKTxAAAACXBIWXMAAA7OAAAOxgEGhk// AAABHElEQVR4nO2TvW3DMBCFHwEPQiGNCw9BwQ0Nt9mAbIK0EjKDIbVBGnoD91JjWEOoSBOI m1xIMfo3HLsMkFcK9929dzytiAiPavUw8VehKiUW57OPEqYpoDjYDLJ03EXQpS+gtiD0YBTn GzzxK5OqtAOKHvAS+wSo1+ALe86SdyTNxwQIVMZILDI5WwefIcGb4vhNAbJnnErP7OEasptE B9nzCYERt6sn0KdHJNYTZxWlLEa/+OQCygQb7LUaVhokWNYYqiONUho02eDirouQz1u/7j7r CKrxZV3/0bSQ1b3bdmIhQOLVQOYa+r2C6mzYI73odj0z290krljRgHZRDJYj/JU+xyVBdBgu YWnPg6QWecaXcCXT/fqHfvQNY+dgcNquFeAAAAAASUVORK5CYII= , обозначается .

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где ϕ i size 12{ϕ rSub { size 8{i} } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAYMCAABS AgAAFgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAIICAABRAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGAAAAGoAAACNAQAAAgDPlSABAAAAAAEA AQDVA4wAAQAAAAAAiwABAAIAAAAfAIoAAQA+AAAAAwA4AAAADABEZWphVnUgU2VyaWYAAAAA AAAZAQAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUA AAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAjQEAABECAAABAGlaAAAAAAABAAEAaQCMAAEA AAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== - однородные полиномы многочлена f size 12{f} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAcUBAADs AQAADwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAMQBAADrAQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBmnAAAAAAAAQAB AGYAjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= .

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B i size 12{A=B rSub { size 8{1} } B rSub { size 8{2} } "." "." "." 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Определение. 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//8CAAAA/////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAUAAAA AAAAAAEAQwBvAG0AcABPAGIAagAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAASAAIA/////wMAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAQAAAGkAAAAAAAAAAwBPAGIAagBJAG4AZgBvAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BAAAAP////8AAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAAAABgAAAAAAAAA= AQAJAAADiwEAAAIAHAAAAAAABQAAAAkCAAAAAAUAAAACAQEAAAAFAAAAAQL///8ABQAAAC4B GAAAAAUAAAALAgAAAAAFAAAADAJAAiAHEgAAACYGDwAaAP////8AABAAAADA////pv///+AG AADmAQAACwAAACYGDwAMAE1hdGhUeXBlAABgABwAAAD7AvP94wAAAAAAkAEAAAACAAIAEFN5 bWJvbAAArhQKQcjwEgBYsfN3YbHzdyBA9XdOEmY7BAAAAC0BAAAIAAAAMgp/AfwAAQAAAFt5 HAAAAPsC8/3jAAAAAACQAQAAAAIAAgAQU3ltYm9sAABiEgonyPASAFix83dhsfN3IED1d04S ZjsEAAAALQEBAAQAAADwAQAACAAAADIKfwF8BgEAAABdeRwAAAD7AiD/AAAAAAAAkAEBAADM BAIAEFRpbWVzIE5ldyBSb21hbgBYsfN3YbHzdyBA9XdOEmY7BAAAAC0BAAAEAAAA8AEBAAgA 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Определение. Полином iVBORw0KGgoAAAANSUhEUgAAAH0AAAAZCAYAAAAc5SFpAAAACXBIWXMAAA7FAAAOxgFsQb88 AAAESElEQVR4nO1aPY6yYBCeN9mDvMRmCw+BscHYegNozLYQz7DB9osN3MCWYGOWQ2xBs5Gb zDfzAgusrPKzuGvkqRTDMPDMzzODT4gIIx4LT7/twIjbYyT9AXGnpCfoLzSwDsUR+w3B1UH8 nk9/C4m/QK30gAzvBKEp1fP5lvTqSQZ4pxBMOexDLV+zEYn2G6Crj0TXQJqhQJM/ReiIGbyX fqslPXIEzrY2vGEIOkTgiAAm8jaOnoCJn8JSH/56j4pz0iOHCOckcolwzjQdXNSbWUt8XGgW HGp+KpeXiyZiOttYgWx2xREdcEZ6FBDjYLfOtLQ0A7UBhLBzG0jw4505nzPpY9keCAXplOGC U1xhCzPBnw1s1Mspw9fUi7kP9+r7yRH2lOjTjfy0W1QObjd59bmMtD2x+x6cQjMNoPz+7kAH DO1/QbruCsSlavrbloaT456IsWGj93GFDcVkx4CVzL7LOawMoGOlm78CfmDBEgGXDojZHo6J CSZQ8ARLQHT/NNmMW/hfLe/Jh1J5dhcVZTz37sOqtVB0zzNDib8Ga9ouAHUXha4+LdGmihUn CfivMWxCs6d3t8Et/P9C+pdMawglvg4H0ITFX2v3uteFXNrPYTqh4KEbXWi4X52I8Ovirx4S nqlKWDMtbTt3pxGG879Cei7i2o5nkr0jxd20BNcjAY4dOMxAyQneDexaOlL1SkymHIAevOg9 zPwahvO/RHqWaV3L9CEm2qB7iY8CUCHHSxnpAwu4/TEB0+xokUTgqwqey34p0QR/UNw19L8L CtIz5Wx47cclOV9RXloQRKSu9W6OJCri8lExFXDW/giJaWJbf9Sadk198OQBaPV+VTaOdjef h8N1//ugRHqsRiO7y+pNmmLn7VGbETdvpDFb78ATPO7LSxkp5isDiXWlXufHBa5h12i5w4gc DeINj48RBPR9G0S80kXeLC7zsW++IzUshSI/rrORbyXPx8R8pPpcFeejZWnEyoOqTstcst3M f6V5wDqk6/HJPwFnI94FfJKeZpoBz7LJYz2H2vVOeJYUXKbPxNxlIZf283KVyauHpYnm27zs QRd7ex1ePAO3FuuE6kOWsqtAHA5N/Y+cNcLuBN56DfG/BcQ84r1wS2zWCjLS80wrxqVOULO+ 2+VE4X79MwdVjxDbjSnFS4bLxxrjG33DY1XlNmt8vXrdGttN/dfdUOj8IoUmpneacEIOkCjG A9ncXbqfDDnpZ5n22IgwUOW7zzQysG0lfCn7M6HLk5dBAdDEpiI98V8rBu4GWzXeqRLR9X26 2jHApnIscmasTQZ5P/9TtlU7tjdZu0onr+lSEpc+kvgF6S+g8j69dO6TIwRu1fvyZnvtvwEp zFAtLDqj+n+BNHhy7VBsxX4eP2M7bcfGqlrMt6Sn3vkemEczhO/ayxP1UtGlC987evX6X8fX oG+XBHf6d6kRfTCS/oAYSX9A/Adyg4KY8CcUPAAAAABJRU5ErkJggg== называется симметрическим, если iVBORw0KGgoAAAANSUhEUgAAAN8AAAAdCAYAAAAq2bmTAAAACXBIWXMAAA7EAAAOxQGMMD9a AAAGgElEQVR4nO1cMY7qMBB9lv5BjLbZYg8RtA0o7d4gNIg2aM+wIu2KJtyAFoVmRQ5BkWZF bjLfjgkhkIQEDITFr/mbj7Ez8zzjGXvMPyKCgYHB7fHv3i9gYPCs0GJ88axPnfkHNoEDDjAd fT4VwjGx7hr+JoDDjf6eBRcaX0hj1oXnrkCBZSbNubAmjEjqkqHvbyhwuNHlE+AC44tp1t8a 3sQY3uWw2IRWwgA76MMY4DPgbOOLZ0MMli5WgaXxdZ4dFhv5PeoMvhE6E7IeIoRX0c/a38A4 jDySdGzwhhVNYBVwWWp86ovL7VPvIB8J6Vt81vOnhZ0anA/ufMIddPE1G8ESCr8XqvlXCMeM up5wwESYmHlwBO4EjByVTnwVpBOFxpcpNRDGFWLMFnjh+w0W8CQh77zo6wYXwWK2C/LuuPqd 5B/KOFWbYq9ukKI8nTg2vnAslAq4q1SpFpIpsN9kIRr0fBjbuw4sYX3wPCxCwYF148Fr8C8s j4ZiVXRXgTG8WhAGuHKJdYeYvQeURhBHxpcYFlzYVllHIakmL+ZY4VrgryKuANa/seCN33To 0/yLNt8DLIXznVa0MTiAZQutepj/xHC26URmfMlZk7d98NBl8u8eHcX68S/W8pNXXjmWCl2Q rJC78790jAfcIb2pPPwFb+IfL4rlg7ZuK1GX/xrO9y9wr1+GbTox/0HsOCT7y4wvOWuys3O7 CxQkX3xhE8geg3Xn+IkdOJhRf2GDaNJ6xR/i9vJwvKqlDzGshCj9YxygLv8nnO9f4P5aMnBJ qrftjx+GnVvFulUxRxxB7oG5hxn4HqwJMdWDTXKpjeIYs68In8KDPCLaKM/OM9fBvveuggb+ 26irpriWDPxFxjPr3fOB8UnF9vDBLxpjf7jEgw+6HbFSk/Aej54jtkceOUFoorlTrfy3R1fn 4xoyLJFmEznjS5PtikWtITgTxk6Aj5Glq8974q/Jk4de/v+Crq4hQw9pxL5nfDH9qmC+OsVv shMXz+grCY0ikbsUbx08VFF2i+TRH3Zq5r+GriQSOdDSTZiaMtTu7nede86ML/7BfCl5etc0 aWKaDUWcvPGBzqD0zEpVAVw+2vVxS3liRCqxKuVCe9hZl/9aO7GndZWroHEvfflroB7fjXqM lrnz8T3jO72RkqDmNng47iD6FHEyD7GQ7RchJhZIVkvYSVWESGL7HQyWxaVLxX2y0qqKdCWQ sbkYh0mv1e+o86jU66eE9wrqEKv6vpY8pah5nKMVdflPt8wrdmJP60p8530KIq44iYpHeiS+ X74Zjo4m8j2Sqk3JnNvO+NSSmMWj5ahWfirwTiniNWWxsDfoguWE5XCmPs2F5vXlmPpxF3lq G4I+1OdfaGDkoycm+nfoSJ1kfdTWldAWb2cRdl0ZwvGQMN3AHw4RffcRyaOJ0Qz9TkmIWlCS uTW+mH7my9olY6r8KTuv2EdR2FUaislJ5toImoS5JTnJURjGHRYcDHoyJCzo++ryFKBOlYle NONf6nbqz6nTHctVYFd/2khXddFSvq1JwCx5o2O5xPpDrKzSUMOIlqLP6dFAop1YFuUKvB8R pcaHqEm+V1Aqcw7kJOu9jgo+kJUEOAgJ1LLtrq6xkaGn72N5tpeNIT2mjUXy916oVPEuPX90 w7rJhvxDTchVJEI3lvB0hQLw9vOtVjPB7dYGEv6FIeb7LL9ylRhfPPvKdXIaOu6dqd21N5uL 8WcUO04WR7+5Rzl4OE6ssWLSng89fR/Lw2e/YmVYCe10oSYpwR4zfFXtFB4Qegs0518hWX1G ItcquTJTa+xkZ+nz6P/bz/c2VHc/t06yiP8+1H2+4itX/8aMURKLbppdDeHOFP68g+7YltvE ZwvgiVmZeIU0bwrkYabwFt4i1y6rOtAPnX3n5JEEyO1qpLmCcC7rHj5GvOTbWXhi3WTVS1fm 5vzvUBDu1UH+vqDMp0D7GyPt51uF6r2PfJCZ5z9AlWr+TYjYeTvWnDnBiiJBHpO23ficRn4/ qRr4IyiRJ5cHxhQt3yCcYwHucSNc3jVLvPLN8ThHTGU45Lv5fL7wB5QUeSPhxVj/lx7ioPzG yOWBMqQUCfkIIcLQIisNe3a/XiYMwfx62dNAy08H6vRi+5UbHlMHOa2sfqiFLA9IIKtDlgN0 mAxDraxZcqPgHu9ncE+07kdzr1IwfDeoUCR7PC8/MvibaJ3xGRg8C4zxGRjcCf8BreMmPXZQ 1B4AAAAASUVORK5CYII= для любой перестановки iVBORw0KGgoAAAANSUhEUgAAAEwAAAAcCAYAAADLGVncAAAACXBIWXMAAA7FAAAOyAHywpKy AAACf0lEQVR4nO1YQY6iQBR9lXiQIr1x4SFgh2HbN4DNxK2mz2Bk3xu4gVsCm4l1CBdsjHWT P7+wFVE6Y40QmwlvJUj9qvfq/1cfJkSEEY9j8uoFDA2jYJYYBbPEKJglRsEsMQpmiVEwS1wE UytBXsw//ATHPIQEBN8kYW4ud6CNK163zH9DH5wm58BZQKBgBeFt8VuHCJHSPAtAtBmcUAZ9 caoEczck3OoyoCVilFojXZf44F0ZKvridONhElMfiDyHM5Z4RzDI7GqiW063gom3GfjlMsHC fSbsT0K3nJqC6ZTWxiRRQpupWgbodE7O9r020Z+OjjldCaYp/cU1fkwAJ0KmNnDd+wEyzAUN xtq653TVVjgoP7jGpULG13GmsHFBK5GxbfJEvD/p3EFU+EiOOT/3WHadjvYldlWM5pjzsW+8 hecSJhvmTKy4agOq3Y8K7gyOyEPZMr49dl+cJucFXRbNYRaJT3HkQTQWIxF+JrR1SrzJR6R6 HfrkNGlLx29TVJcolgFyW+/yp63eYY5+Xvv1xCK/mfiv5dISu09OVq9GKot5fYuWP0z3jJbS UJRVJdfHAdFN7HtOikvW487NZGKArPpd24aFYJoOe2AWSE75lHQY1h4wW3L4lsWsKhW/yqJb dBP7npNMD+xvO94CD54wUxCClcD6oLkkpP3Ld8xR9saAzx6Qm2aQdyXO7p6tu+3u0WXsBicj mmlFcPY6Toq9j/eFrJ61EEyKkzj/E77h1PA1TWUxAydhhfHzTgsavqYyxHywLKCglEtPC3b5 hMKIxampGuKnoBq1r1WQU/hFBEeYEnWfz7C71mDwOJVpfdlsdcaStMQomCVGwSzxB/lAoLib A5cqAAAAAElFTkSuQmCC переменных iVBORw0KGgoAAAANSUhEUgAAAEQAAAAZCAYAAACIA4ibAAAACXBIWXMAAA7GAAAOxgHq1c2S AAACP0lEQVR4nO1XMW6rQBB9K/kgi9KkyCFAaUC0uQFuLLdGPkMEbeQGbpAWQROFQ7ig+fLe ZDJLjI3XWLZj8k0kXuVds29nHjtvlgkRYcQek3sHMDSMghgYBTEwCmJgFMTAKIiBURADoyAG doKUoSAn5h9ugk0eQAKCJ0noycUnKLLF/cK8DH3kMGmIMp9AfgjhvONDBQiQkpf5IIoGL4RG XznUgtgRCbse+rRAjEoppK8VlqzyX0FfORgeIvHoAlPH4hNGrDD+xOk4xG05mIKIhyfw116C ud1jjP8Vt+VwKIhK6VWbEiooTX1iUW1eGKjRXpjDKbQEUZTOuOY2CWBNkZURbNvcyyNrWnwP Fj8M+FdxPodzaLVdC9WSa06WyHgcZyUiGxSKjG2KiXUtPq/YsaWoham6Cb9b3wKfzZqj/1DX NnML/TY9DrxotclGdDfZIA9kx/pu7styYKP1LEwLF8kmx8ObgNmmJ00AuyB52TxxKZ46EMbm UsrhlQj2J/dcDmU4I6w2SGYzVG8eKt2m5yk8a19eExnkgozO1DV3FdzHztrVrZFja28kcmOj s3t3cF+agx3lwkZJYVFg/cInUItXVlQw52r7TM9X95KyuiS2t8QhcpcZ31L41ARyO4zhsjgN 548EUZU21uXxXqED7I5tv+iLW/1bs6rLrQ0o0sMnX3LZpaSC4DpBDroMdH2C2ua3vy32j364 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Группа перестановок из n size 12{n} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAARMCAADs AQAADwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAABICAADrAQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBuEAEAAAAAAQAB AG4AjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= - элементов реализуется в виде конечной подгруппы матриц n size 12{n} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAARMCAADs AQAADwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAABICAADrAQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBuEAEAAAAAAQAB AG4AjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= - ого порядка, которые получаются всевозможными перестановками столбцов единичной матрицы.

Теорема (основное свойство симметрических полиномов). Каждый симметрический полином из iVBORw0KGgoAAAANSUhEUgAAAFwAAAAZCAYAAAC8ekmHAAAACXBIWXMAAA7FAAAOxgFsQb88 AAADRElEQVR4nO1ZPW6zQBB9K/kgi9Kk8CFANFhufYOlsdIa+QyR3UZuzA3cItxY4RAuaD6Z m8w3u0CME5Jg/IOT+DUxKzQ7+2Z585MeEeGO66HXtQN/DXfCr4w74VdGC8IzCgcW/PV+ZfJK mNkQ53Pr5yMLB2RVSPKWO8RKil7dC40InLyCZvad5E8gVSxI6V8JBcLBtljvVV/YQZPex9Du wsW/gQNJyVK+4d4IsiNn/gIqhGf0b6v5djXhd6m4EPaEZxus+IL3p7J4Dmlg+ShUHa80g90g EEkgyJlDZwnsYpUHLwlI6MUfoPuX9r9CeMrkehjJ4lm6GHngtcrGDZyNhgQaBhDOCptMQYED Fw1BNLtpojWu4f8b4Uk0N1F1Zf6chWP4/eMias9I2ObXkCaYI80yhM8pprE6h68XxzX8LwjP 9Rv9B77JvMnAotVox2TLllGVeOSvw3csU2KqH5cTLud/STh0gYK1A6H1i6VluZAnmJXioQ8C y9GTfaqLXeBy/ueEJxE0z6bhkSF0slxtMigl21nlhPtsApdyKPFpmWkSFG4wkTb0vw0M4ZnR k0nR8OTJ0l9tkClF8ujPiVv/Mevebglw4KKEqxv73RvVtndy2gHOj+/9PwU9vcFmVW14pHBH HjHjJku7mwGNsTBzgCYGk8BCOmXdkwkifp5HiR4TcHsbcSoqSkt3wVlfCkN8WmdDl2b1pWhZ tr2NH8rytVLGlQEt5xdNbTfz3+Q4+GuW3V2MhxeBD2XkN4Qb/faW+4ZHuiNWcR++JWqdrkN5 yP0cxsbT0qO5r/PC4QGlbJuML4em/ifBmLDYYTkeI30ZINVl5JOW4Wbyw4TbYvb+vz5SiZiO K4X2w5qv1xrDe6x1XpdufPbqJh98/XbfGttN/bdnsbD1QGq9xpYruVgHJ0lpzTYXX52nwA3O wxOKjGQ0a7Y6sW2KDL71RVGhexiPyW9isz3hc1NCmk+j7TzcDMswPVhLAge40Hz9XLZNkTGZ FhKZ9zD9IXcwYUhcaJj1D/Pw4m8LwqVQsWkG2jt84EweuDJX7Lu98+M8tvMiwxsdCsjcEdjq M5R56hNZ60RSTtL2zvH+wh13AW9Qw3837oRfGf8BouvvHHIwLKoAAAAASUVORK5CYII= может быть однозначно представлен в виде полинома от элементарных симметрических функций iVBORw0KGgoAAAANSUhEUgAAAF0AAAAZCAYAAABTuCK5AAAACXBIWXMAAA7EAAAOxgGnHWyZ AAACyUlEQVR4nO1ZQY6qQBB9nXiQNm5czCEwbjBuvQFujFuIZzC6NW7kBrM1sJnIIWbBZmLf pH614P+g8EVp4Seft5mAdL2uR3W90ukRETo0i17bG/gf0YneAjrRW0AnegvoRG8BnegtoBO9 BXSit4A70SNP0Ghb8KR9wDlwIAFRl7QJjnejTg5/RI88EjqKewKRJaB8mvRjrGgDy5QITXC8 GwZySETXCzmQfTgjcGSyUA7wgTnW/hKWI+tvtgmOd8NQDhfRo90coT4WBYvCWOnItffbBMe7 YSoHFj2i41a3onG+D6kffJvY6QVNcLwb5nLoXRd9DGT+ExUj5D/u1Lqn1yYC7mkbq1offooj Ik+M8Nuj3Cd43okXdCpDycioyF/rtF1kYyl/Qv15mFy4lTlKUMShBV9jeCaQ5Gq6mNYa/tIi R/6LRlus0yP0IMeY2cD8GGFjJSsjrw+trXu6ceTxnh1biov4cXHAZJRyccq6eWUOS2wos3s5 hF1weAs5cp/puMRcSKeLtBeno9y1eHKG+Ch25Rz4RUz0fRuHc4DBTuAyWmb4udKlcIITxXyk xRbpfzSSBbfVJaV8sdqqc2Shvj6Bw/6vzzSHajlE3gLYn3FYLBDvJoinfGqXPvRYqZBYbdpe dIVxZZjanz0s8PEnObi19OMV9/OSF13IwSwbEpQlkY4IyMk9I51A3NyqFLtKDtYmEJZuk2GI 7xmfJH3aophCjrlPnzH8M0Di8O6p3rfKpH2tSgzUDEcxDMWOjjwIcItKR8uIg9r8Aq4xXxJd xdpMV/dc3gi49tKXN+zlBedrj2vrGtMIRym1mdjqh33IXaV9XpG+/JhKLiaflOM8J3puekHS 27JmpI+2VWe3+lheXGeb6ZuJUV1Rn6McZmIr+vpkk57tc3e3I4FvrVVipNXxsBfWhmFvaQXa cAlO6XX3024r6ERvAb8AdHHOJMv5KyMAAAAASUVORK5CYII= .

Определение. Пусть 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACQAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAAYAAAD+//// //////////8BAAAA//////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// 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Предложение. 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HAAAAPsCEAAHAAAAAAC8AgAAAMwBAgIiU3lzdGVtAMorE2bKAAAKACEAigEAAAAAAAAAAFTz EgAEAAAALQEAAAQAAADwAQEAAwAAAAAA - оператор Рейнольдса конечной матричной группы 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACAAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAP7///////// /////wEAAAD///////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////wMA TwBDAFgATgBBAE0ARQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAASAAIA/////wUAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA/v///wAAAAAAAAAAYwBvAG4AdABlAG4AdABzAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BgAAAP////8AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD+////AAAAAAAAAABFAHEAdQBhAHQAaQBvAG4A IABOAGEAdABpAHYAZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAIAACAP// /////////////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAADlAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA////////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA/v///wAAAAAAAAAA/v///wIAAAD+/////v///wUAAAAGAAAABwAAAAgA AAAJAAAACgAAAAsAAAAMAAAADQAAAA4AAAAPAAAAEAAAABEAAAD+////EwAAABQAAAAVAAAA /v////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////8BAAAC CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAQD+/wMKAAD/////A84CAAAAAADAAAAAAAAARhYAAABNYXRoVHlwZSA1LjAgRXF1 YXRpb24ADAAAAE1hdGhUeXBlIEVGAA8AAABFcXVhdGlvbi5EU01UNAD0ObJxAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAMAAQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA/////wMAAAAEAAAAAQAAAP// //8AAAAAAAAAAKkBAAD3AQAARgMAAAEACQAAA6MBAAAHABwAAAAAAAQAAAADAQgABQAAAAsC AAAAAAUAAAAMAjwAMwAEAAAALgEYABwAAAD7AhAABwAAAAAAvAIAAADMAQICIlN5c3RlbQAA MwAAACo+AACcWREABO6DOfDoHQAMAgAABAAAAC0BAAAEAAAAAgEBAAUAAAAJAgAAAAIcAAAA +wLG/wAAAAAAAJABAAAAzARAABJUaW1lcyBOZXcgUm9tYW4AAAAAAAAAAAAAAAAAAAAAAAQA AAAtAQEAAwAAAB4ABwAAABYEOwAyAAAAAAADAAAAHgAEAAAALAEAAAMAAAAeAAMAAAAeAAgA AAD6AgAAAAAAAAAAAAAEAAAALQECAAcAAAD8AgAA////AAAABAAAAC0BAwAEAAAALgEAAAUA AAAJAgAAAAAFAAAAAQL///8ABAAAAAQBDQAcAAAA+wL4/wAAAAAAAJABAAAAzAdAABJUaW1l 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//8CAAAA/////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAUAAAA AAAAAAEAQwBvAG0AcABPAGIAagAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAASAAIA/////wMAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAQAAAGkAAAAAAAAAAwBPAGIAagBJAG4AZgBvAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BAAAAP////8AAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAAAABgAAAAAAAAA= AQAJAAADnQAAAAIAHAAAAAAABQAAAAkCAAAAAAUAAAACAQEAAAAFAAAAAQL///8ABQAAAC4B GAAAAAUAAAALAgAAAAAFAAAADALAAaABEgAAACYGDwAaAP////8AABAAAADA////xv///2AB AACGAQAACwAAACYGDwAMAE1hdGhUeXBlAAAwABwAAAD7AoD+AAAAAAAAkAEBAADMBAIAEFRp bWVzIE5ldyBSb21hbgBYsfN3YbHzdyBA9XfbEmZQBAAAAC0BAAAIAAAAMgpgAS4AAQAAAEd5 CgAAACYGDwAKAP////8BAAAAAAAcAAAA+wIQAAcAAAAAALwCAAAAzAECAiJTeXN0ZW0AANsS ZlAAAAoAIQCKAQAAAAD/////VPMSAAQAAAAtAQEABAAAAPABAAADAAAAAAA= . Тогда:

• 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAABwAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAA/v////////////// AQAAAP////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////wMA TwBDAFgATgBBAE0ARQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAASAAIA/////wUAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA/v///wAAAAAAAAAAYwBvAG4AdABlAG4AdABzAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BgAAAP////8AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD+////AAAAAAAAAABFAHEAdQBhAHQAaQBvAG4A IABOAGEAdABpAHYAZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAIAACAP// /////////////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAsAAAD2AAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA////////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA/v///wAAAAAAAAAA/v///wIAAAD+/////v///wUAAAAGAAAABwAAAAgA AAAJAAAACgAAAP7///8MAAAADQAAAA4AAAD+//////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////8BAAAC CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 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AAAAAAAAAAAAABIAAgD/////BAAAAP////8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAADAAAABgAAAAAAAAADAE8AQwBYAE4AQQBNAEUAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEgACAP////8FAAAA/////wAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAP7///8AAAAAAAAAAGMAbwBuAHQAZQBuAHQA cwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAASAAIA /////wYAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA/v///wAA AAAAAAAAAgBPAGwAZQBQAHIAZQBzADAAMAAwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAABgAAgD/////BwAAAP////8AAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAEAAAAugEAAAAAAABFAHEAdQBhAHQAaQBvAG4AIABOAGEAdABpAHYA ZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAIAACAP///////////////wAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAsAAAD2AAAAAAAAAFIAbwBvAHQA IABFAG4AdAByAHkAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAWAAUA//////////8BAAAAA84CAAAAAADAAAAAAAAARgAAAAAAAAAAAAAAAAAAAAAAAAAA AwAAAMADAAAAAAAAAQBPAGwAZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAoAAgD/////AgAAAP////8AAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAFAAAAAAAAAABAEMAbwBtAHAATwBiAGoAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEgACAP////8DAAAA /////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAABpAAAAAAAAAAMA TwBiAGoASQBuAGYAbwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAASAAIA/////wQAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAwAAAAYAAAAAAAAA AQAJAAADyQAAAAIAHAAAAAAABQAAAAkCAAAAAAUAAAACAQEAAAAFAAAAAQL///8ABQAAAC4B GAAAAAUAAAALAgAAAAAFAAAADAJAAiACEgAAACYGDwAaAP////8AABAAAADA////pv///+AB AADmAQAACwAAACYGDwAMAE1hdGhUeXBlAABgABwAAAD7AiD/AAAAAAAAkAEBAADMBAIAEFRp bWVzIE5ldyBSb21hbgBYsfN3YbHzdyBA9XciE2ZSBAAAAC0BAAAIAAAAMgrgAS8BAQAAAEd5 HAAAAPsCgP4AAAAAAACQAQEAAMwEAgAQVGltZXMgTmV3IFJvbWFuAFix83dhsfN3IED1dyIT ZlIEAAAALQEBAAQAAADwAQAACAAAADIKgAFGAAEAAABSeQoAAAAmBg8ACgD/////AQAAAAAA HAAAAPsCEAAHAAAAAAC8AgAAAMwBAgIiU3lzdGVtAFIiE2ZSAAAKACEAigEAAAAAAAAAALTz EgAEAAAALQEAAAQAAADwAQEAAwAAAAAA является линейным оператором;
• Если 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACgAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAAYAAAAHAAAA /v//////////////AQAAAP////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////wMA TwBDAFgATgBBAE0ARQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAASAAIA/////wUAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA/v///wAAAAAAAAAAYwBvAG4AdABlAG4AdABzAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BgAAAP////8AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD+////AAAAAAAAAABFAHEAdQBhAHQAaQBvAG4A IABOAGEAdABpAHYAZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAIAACAP// /////////////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAB8AAABiAQAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA////////////////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAA/v///wAAAAAAAAAA/v///wIAAAD+/////v///wUAAAAGAAAABwAAAAgA AAAJAAAACgAAAAsAAAAMAAAADQAAAA4AAAAPAAAAEAAAABEAAAASAAAAEwAAABQAAAAVAAAA FgAAABcAAAAYAAAAGQAAABoAAAAbAAAAHAAAAB0AAAAeAAAA/v///yAAAAAhAAAAIgAAACMA AAAkAAAA/v////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////8BAAAC CAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 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• Многочлен инвариантен тогда и только тогда, когда инвариантны все его однородные составляющие;
• Оператор Рейнольдса переводит любой однородный многочлен в однородный многочлен той же степени.

Следствием последнего свойства является метод построения инвариантов произвольной конечной матричной группы iVBORw0KGgoAAAANSUhEUgAAAA0AAAAZCAYAAADqrKTxAAAACXBIWXMAAA7OAAAOxgEGhk// AAABHElEQVR4nO2TvW3DMBCFHwEPQiGNCw9BwQ0Nt9mAbIK0EjKDIbVBGnoD91JjWEOoSBOI m1xIMfo3HLsMkFcK9929dzytiAiPavUw8VehKiUW57OPEqYpoDjYDLJ03EXQpS+gtiD0YBTn GzzxK5OqtAOKHvAS+wSo1+ALe86SdyTNxwQIVMZILDI5WwefIcGb4vhNAbJnnErP7OEasptE B9nzCYERt6sn0KdHJNYTZxWlLEa/+OQCygQb7LUaVhokWNYYqiONUho02eDirouQz1u/7j7r CKrxZV3/0bSQ1b3bdmIhQOLVQOYa+r2C6mzYI73odj0z290krljRgHZRDJYj/JU+xyVBdBgu YWnPg6QWecaXcCXT/fqHfvQNY+dgcNquFeAAAAAASUVORK5CYII= .

Следующая теорема, принадлежащая Эмми Нётер, утверждает, что с помощью оператора Рейнольдса всегда можно построить конечное число инвариантов, порождающих кольцо 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACAAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAP7///////// /////wEAAAD///////////////////////////////////////////////////////////// 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Теорема. 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k [ x 1 , . . . , x n ] G = k [ R G ( x β ) : | β | ≤ | G | ] size 12{k left [ size 10{x rSub {1} size 10{, "." "." "." ,x rSub {n} }} right ] rSup {G} =k left [R rSub {G} left (x rSup {β} right ): lline β rline ,

где x β size 12{x rSup {β} } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAZUCAAAi AgAAFgAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAJQCAAAhAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAADCAQAAAQB47gAAAAAAAQAB AHgAjAABAAAAAACLAAEAAgAAAB8AigABAD4AAAADADgAAAAMAERlamFWdSBTZXJpZgAAAAAA AP0AAAAAAAAAAAAFAAAAAAACAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAA AP////8AhgABAAQAAAAAAAAAcgACABgAAAByAQAA7gAAAAIAzrKSAAAAAAABAAEAsgOMAAEA AAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA== - любые мономы с полной степенью β size 12{β} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAegBAADs AQAADwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAOcBAADrAQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAGAAAAGoAAACNAQAAAgDOsvQAAAAAAAEA AQCyA4wAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA , удовлетворяющей условию | β | ≤ | G | size 12{ lline β rline .

В частности, кольцо 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACQAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAAYAAAD+//// //////////8BAAAA//////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////wMA TwBDAFgATgBBAE0ARQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAASAAIA/////wUAAAD/////AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAA/v///wAAAAAAAAAAYwBvAG4AdABlAG4AdABzAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABIAAgD/////BgAAAP////8AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD+////AAAAAAAAAABFAHEAdQBhAHQAaQBvAG4A IABOAGEAdABpAHYAZQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAIAACAP// /////////////wAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABoAAABQAQAA 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Приведённая теорема позволяет найти все инварианты, порождающие кольцо k [ x 1 , . . . , x n ] G size 12{k left [x rSub { size 8{1} } , "." "." "." ,x rSub { size 8{n} } right ] rSup { size 8{G} } } {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAATQLAADl AgAAYwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAADMLAADkAgAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAD3AQAAAQBrAAEAAAAAAQAB AGsAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAkBAABp AgAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAaAAAApwEAAGECAAABAAAAWwBLAAAAAAABAAEAWwCMAAEA AAAAAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNlcmlmAAAAAAAApgEAAAAA AAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCG AAEABAAAAAAAAAByAAIAFwAAACwCAAD3AQAAAQB47gAAAAAAAQABAHgAjAABAAAAAACLAAEA AgAAAB8AigABAD4AAAADADgAAAAMAERlamFWdSBTZXJpZgAAAAAAABkBAAAAAAAAAAAFAAAA AAAAAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA AAAAcgACABcAAAA0AwAAewIAAAEAMbMAAAAAAAEAAQAxAIwAAQAAAAAAiwABAAIAAAAfAIoA AQA+AAAAAwA4AAAADABEZWphVnUgU2VyaWYAAAAAAACmAQAAAAAAAAAABQAAAAAAAgD/AwAA AAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAX AAAACAQAAPcBAAABACyGAAAAAAABAAEALACMAAEAAAAAAIsAAQACAAAAHwCKAAEAPgAAAAMA OAAAAAwARGVqYVZ1IFNlcmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAAA/wMAAAAAAAAAAQD/ AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAMEEAAD3 AQAAAQAuhgAAAAAAAQABAC4AjAABAAAAAACLAAEAAgAAAB8AigABAD4AAAADADgAAAAMAERl amFWdSBTZXJpZgAAAAAAAKYBAAAAAAAAAAAFAAAAAAAAAP8DAAAAAAAAAAEA/wMAAAAAAIgA AQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAAB6BQAA9wEAAAEALoYA AAAAAAEAAQAuAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA+AAAAAwA4AAAADABEZWphVnUgU2Vy aWYAAAAAAACmAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAGQYAAPcBAAABAC6GAAAAAAABAAEA LgCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNlcmlmAAAAAAAA pgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA /////wCGAAEABAAAAAAAAAByAAIAFwAAANIGAAD3AQAAAQAshgAAAAAAAQABACwAjAABAAAA AACLAAEAAgAAAB8AigABAD4AAAADADgAAAAMAERlamFWdSBTZXJpZgAAAAAAAKYBAAAAAAAA AAAFAAAAAAACAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgAB AAQAAAAAAAAAcgACABcAAACLBwAA9wEAAAEAeO4AAAAAAAEAAQB4AIwAAQAAAAAAiwABAAIA AAAfAIoAAQA+AAAAAwA4AAAADABEZWphVnUgU2VyaWYAAAAAAAAZAQAAAAAAAAAABQAAAAAA AgD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAA AHIAAgAXAAAAlAgAAHsCAAABAG61AAAAAAABAAEAbgCMAAEAAAAAAIsAAQACAAAAHwCKAAEA PAAAAAMANgAAAAoAT3BlblN5bWJvbAAACQEAAGkCAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAA AAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAACC CQAAYQIAAAEAAABdAEsAAAAAAAEAAQBdAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA+AAAAAwA4 AAAADABEZWphVnUgU2VyaWYAAAAAAAAZAQAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAABAP8D AAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAtwkAAAkB AAABAEfhAAAAAAABAAEARwCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAA AA== .

Эмми Нётер в молодости и в зрелости

Рассмотрим пример построения множества образующих кольца инвариантов для циклической группы iVBORw0KGgoAAAANSUhEUgAAAG8AAAAZCAYAAAA/vnC8AAAACXBIWXMAAA7FAAAOxgFsQb88 AAAFZElEQVR4nO1aO47qShA9Ld2F2CIhYBG2SIxIZwd2gkhtsQZkp4jE7IB0ZJIRXgQBCcI7 6VfV7T8/85k7b644Caan3V1dp77N/JFS4o3fiT8/LcAbj+NN3i/Gm7xfjDd5F5HKQNjYxUck riF+QoJsNZKmN8BWhrCAExnOk5cGUthRa9C/uMjPIJOrkQlv0xx1Gspuz+l2hjQQ0o54rkTY mKsJrWvG6UTueVnbaK9luImQLu8pMI+Psr1Pi7xiEwc0F67Bgudj6MO4vvffQ7aSI9PDxolx lC7LVcnpLZC6obTUmCHchA6iztSNOLZ2TVxzrvaCDfytBC2uxhXJnokRThXbBMuxlXsxR/+Y wM0CiHkfx4RlTxGMDpjScxYIiFEs9XixtyVCuSUCT/dpkJcGBXFJTpzeePjhSOx7tQV/EkTS hIhjL2ockhS0jOXa3D+x9EpOFEHJeZL9LUKrGremMZyIZFl/IXNdaVzVj4G+c27cEtOPuZys hkjCI+LRBF+Zy/pvzAm3PkXDCVbDRBbcVORRqORI6cTLGnH5tuy+Hc5+Czoc3ZjE3tQgpb3I QoUfJ56eKthwRfKEoOlCe/PSOv3bWR0YPQzo40Y0vAmjN8DmMwOT1BtspHo0WpOsMZlrhPVX BjdnNiePrHnOWvUxc9tvvQJFOH4+b6afmv1Bz3iRbOXKUi3t3xFhsgN2WpinolJ22MHpT/lJ HnYO1OMJLDH2IaOal2vysi+s2Xz88bcUJNlqkofjZwsePhx/0gGN18hWLa2JcO5YOPtak9fR 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В работе также построены инварианты следующих групп кольца многочленов iVBORw0KGgoAAAANSUhEUgAAADcAAAAZCAYAAACVfbYAAAAACXBIWXMAAA7EAAAOxgGnHWyZ AAADOUlEQVR4nO1YO46jQBSsluYgjZw4mEOAJgGRzg0gsUhBPgOC1JoEbrApwskKDuGAxIKb vH3Nx/aOwODx/Ha0lUy7oZtX3a+qX88DEeGn4uGrA/hI/Cf3r+JOcg2llgZ3f+7xC0KkQ9w3 741RpBZpF0GYSY3ckeJhLMAxDANGH/oFKNI/ldAlpJMLclSrpEAYOPT9TE4KJy+oEiHWdQ6n CSDCNercgUSJwDrC43YTCAgroa7/c3fmrejTUmJtjj3Whfcc0iZ9Qh7VSKwNfjcOHPmJEd6B Wc3J1SP2WQNFdPW4p7YpPziqd8IsueZ4gLn2VIuOBxNtcyHKQJARQwkWp3QuAxKqc4lOh3dh IlGSkb0cTv3XDWyGXEk7F3iupbIkVM8viOQyvSlimU0gmzVs/OrSGSlZmQ2iaH4ORaB914Yy CXdXwmEWLfRIFH7MC+fD1qenGCe3d6EJt22qlTmbjFzCq/8+ie67NvmIUTUN0rDCNneWTiCo D9z2QfHhiAY6yQszMxMP+hVzGyen0ugF2OxW8I4WhDonuO9t6MzKNbRuod7gtFJNEFdMbpB7 SRnv2pbk1XHTaSkdbDmODWrUyQaa66J1zZutUrIRgavzBJ5+49BhBp4AfHodG95Q9fkyw4F3 LZpZqKua070EobZjK8lRVAKGu0H6lJOzUHctmpTCVvuXK38j5JotRaW2anPREYLTe36m64Yi HbH1BYWpx+dcAT9+JexZcCAb1lnNKa25yMoI+tKhf8WxQrt3auuwQ7WNFqV3T65BxbJaD737 8yoPu1c6EaKCrcEwENjsdwvqxzLQOBDWmSyR8e84K5Vtc4mUsc1ErRl0x4WPov89wQ6d7HYI KhtRtCxz+trSQJs5moDbPzCstgQj2e+eIeLToNiwVKlGU/IbCtnzGaTDS0yKXQNilsgouU63 sfJeffGovrZsXWwSytY5njGMXuPPhez1vhPM9SIt+sVti/LF9zll6SromWKcD/RwzbeSG69S 70MuVunW7eIt97kyMICJ98vAoqPX31JUpRJNXLcwcp/r/95Jbj6lr+FcxbyGqmP3cJUHLKhB p1L+m/6b4b5FG/BNyb0PfjS5PxwLhblmgWmnAAAAAElFTkSuQmCC :

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2) четверная группа Клейна:

3) циклическая группа iVBORw0KGgoAAAANSUhEUgAAANQAAAApCAYAAABN0gffAAAACXBIWXMAAA7EAAAOxgGnHWyZ AAAHoUlEQVR4nO2dP276MBTHn6XfQRJ16dBDGLGk6tobJEvFGtQzIFirLnAD1goWlByiA0tF buKf/wVC6jhOiPOH5rNUQEjMN3n28/Pz6z9CCIyMjDTDv64bMDJyT4wGBTGZowl8r0+w8x1k +2rJ5pm4wf78OowILDFYv6592tVRR5ca/2GDSsjm2YVUd6+FK8ZzRCarECKyAwyA+OsJvc8R IcM1qvZ11NG1xoM3KCEgvZElPSM/DiKgqiL6gqDJN6xPBEgyB8ROYJtkQxa8nTN+o9lbeBlB uJrAarGBGfaJw97nbWP3f3k+rmmutLjtRO3rqMNUYx0G+p9HwPC3hgM2KOFirLw1nIgPTqFQ sgd9yvx4vETnWEwCrURlksMW9rT/Xk+dzLsOPLIufX+kzWCvZNsielPRM31Qd8R3mjQqhRa3 0IGOOow11mGgv+PvEPGFlgh5hB4H6XGDNah4Lo1ppzMmdlyDD9ANJEfmE4Xw4GTfddDDE30Q 99/wk9B7mX5Gb+ppTXtBdw4PZElwQyNVX7SwRSWNdRjp7yB/R4dm1kG9bWC6E6PfMA3qPLRP tcbEhm/hT+PWmqYmIT/f6k8c0X3CMYGr7pP1gtGRumbzF2YAtzehN1rYorrGOsz0p0b1HpJg soVD4rNRaqAjVHJUDO15qEtIfXo2t8KW5iK2wbM1eO4E5i+3TqiHr0UXGOnvPNIncQXbQwI+ tahBGlRS1BVlj9ks6M+kPTLrNoaKM4VX2rkGphPqAsq0SAM7Rhi42XeDif7OAzzRP+kTOUiD EjzlfOUsCTlsWRTmvSc9curHF33uwaOj/l7epahOuRZ4SRBZ1jl3n6irsf6cpvrvpT85YIPS TDKTA4hnCLfbJA3Cj//dZvVEOgN+oZ9eXIrK9FALW9TWWEdF/dUGxWPxeR8gtLo20iRp+PTV 6bolF5wH5hjkb0xMvpjM4YtGVxH2XW0PkPjV3b4+amGL+hprz6rXP/nh7p4nh7+cQaWr3h5f rBOxdfkePJoGSKyTCqeO2kgXp26PZAs8g7W3giB4g81UrG+IuQ3VeoY1X5SuzMpwHeWKnmph i9oa6yjRnwfIqMJS4CuD4usU3JguC1XshNNXj8DxoT8TUR5ZoYO70udLgI/wnq4DuE6XYewD F1DAFidtjcSZdQsX0Q7K/FrClVnBV0yPxVWuaaLFLXSho476GmvPqtFfBMgu87OLQfF1ChbE +YT86jBfGa7bGhvkIitXyCEYnnQdABOejsC22ldIveuKEXlf0IFoMNLiFrrSUUfzbaqivzQo 2tMsuKMJ70MOM2fwqod0+oscketyV1p0Qan+l4izMCgZCao/cWsZ2fM+qSYG0qftOuvZBvsq S/2MO9aiC1T6i9HrElmUBiUnVi/YSkOaXjiMPwLYhxHsBrvlYeRuwEsUhfT5lvl83KBiHles s/Bles2GFg6TDXl2hTGVJXgqR68/yqiFXdjzHdFBw0XbbJRPl3nQExwf7YjPRzw05zvGxhFq pHO4B/YtthFVzJSQe5DSlwYjxfmCDbp8JkmLlSNid8yohUV4dJwtNYlnNmNQilQe5mK9AXzy BzyGOVrAI9ud6aQ7SxewmeHSTXCN54qlYXPVg3JjRKzPVI7W3bEWXaDSX6xD5aJ8vMdfBRB8 xOCnez8y8xWHjxYYltm9NPxmlWd9d0XliFifKYnWXeooqBcw70qLLiiNluajfGxucgJqQBNA q+utzEW9IssRA8UicCukC5Yq5OhVmHScPU1n1XGqVQhKt6tUDi5U0KK0DXdVrcme/heXT074 zdozJ+7xnc6fOioXlcufukbWEPj+gQRwYTJpN9Vx6lUIEtnSJVFYZXqRmRZldF1JqDma1z+f V1p5+wbvqbgxdRdhy+dPXSNzDwPNHpYmquNUpXaFILm123sF9QZlkU0dRqpAjoEWpZfvQCsb 2NI/l1dazaDOI5M0Jvp6DsveDf3O9JX+yKCwhkAj1XGqUrtCkEhwLaqfwYrVgMb9KtOi9Opd aGUDS/rnqWBQoi4BG96y86ww6mqrp2bdLN26/BXTBw3/+rix6jhtEH+J7QcF9TNYBBXrvl+i RRmD0soGJfqn89RV9R27GC2pifdnp7TuZqauzhfES5wrA9VsdRzb8CwWb13g7pmg06KMYWll A1P9CzYYDoM0IVF7jP8OYTCBxYb6/kPNoJfzF/X8yJy70KILTPTPRZwHaVD50k1qMJqtPeIG HxD7zRWLbBOeBEx7x09865mGr0VKmxWajPTPlbQbqEHJeUFJnQXH/4T11oXrYoU2quNYIJfS cuvp1FqUfqt3WrVWoclIf7mPMOMSDtOg2K7MzzXZugG4z8B6oQKjYrs3I3JEE3h+PJF0Ec9K dZwmYVkqPMS7a3DhXK1F6bf6rpUNjPTP1l+5GN1ADQrkQvSU/ygXBawqasFDwoIpJ1HY/Sgy 1O1Ux2mI839/2Flox28tyr7Ra61sYKD/OWtE8Y8qhmtQHNP6AbnjrFTHaYjseokVKtZc6LNW NjDQX/z3DfVnAzeoutipjqOnbxWCTOlCKxu0o/8fNShG2xV7+lghyJQhtz2lnd/whw1qZKR5 RoMaGWmQ0aBGRhrkPzkakP2xmob8AAAAAElFTkSuQmCC второго порядка. iVBORw0KGgoAAAANSUhEUgAAAL0AAAAaCAYAAADizl1mAAAACXBIWXMAAA7EAAAOyQEghXBW AAAFLUlEQVR4nO1bO47yMBCeSBzEaBsKDhFEkxUtN0iaFS0RZ0BJu9oGbkCLQoM2h9giDcI3 8T92yINnHCcx4SdftUDieX0ez9jeHmMMOnR4J/SerUCHDrrRkb7D2+G/Jz1df7K+s8u+mP8C 80zjeRo9AyFzjRH4ycdGfaBTlhoUSE/Z+rMP5zxi4Jmg1bBLMlurIwQ2OdMhdA028ufwywIw AQz+zhcQnWrWgKr+5iRcwuDIgBF8J3SZMVrCemYym9QdM52yinGPI71bD0g59ckzmNiBwWz+ V5xZ/i5+5/bEhPcE4ZN3At2K3oE+f5uGx8zsIxmAdeWtuqBTVjHucaSXf+AIPBBDmJjPULFO ULbf7HBm/6SEbxue5W+63wCgX3RkXp2yyuCsvKERZh5r+nIFwDUocFOGC/JsRR5Cu7+x3OhH C1wxSPMk1CmrJHKkp+zwx2Mw5kFonaKlQA+4lFkwJbd+vGi04LwE0ge9/hbllCBh8yWpTlkq yEhP97DJZ0e6Zp99B05VpzQx4uYReNcAx8COAyoaGl+uLk2eRdKujkG2NKbfS9TA5AOGqPlm T8G2SarXdiLey9lDgTeJy/UMzNNz2qDT32nWNVNfuuDJNcNl41FFVgl7qnAkR/oIHZ7LjmQM Uwvwu5xwCYU5sdjEBWO0gT21wQYM5nYCjHlyDhbPToBnY+c7BDtmKfZInvE790VzWlwDY0N1 XCGJ+mA4cDpyRgfNTr+lzRZl0W4Oi4DcHKVRaPM3rmyCCD4YPqTH7/Nfr1jH0vGoIEvWnho4 kpI+3Ppido1J/Jmuv8AZltsxMD1mxLImbI6GRxQz6TJCUtmyAxgJHydzYP7fAXOxyUiOABYy 15QpB4htBOyBXJFZUTeJjJpmHxlYcqTV528+yTHryQ56LqBkPCrIAkl7auDIifRxfQnDDwyW WPLZZnqs0IQQGGDWckZ9sczYCjUr4QP4EWoDp0YvZFucwQtG1FTKg2eL5QCOSHhC1+Dux8yz 79vKg8FUI3kT7fN3oYQm43EtTcoeVZ0S0ovdDtiN+LIEohT4efxigdLGx5AvbyuYmYoj4ACA 7eiBIum4KuEW/nAGe5UDmizBAH2sfTj4oYVetM/fhRIai8dNaVL2qOoUkx4f5r4XxT9ZA2+o 8k1gaWDpsBTBzM/CkhAHG3yJ439Ttl5CTbV3+SW49vKmjf4uQmPxuAFZexR1EqSnYq1Niv+4 oXI2e6C2fVYrSWrM1l9Yhx1XmEod2IZYM5vlRhAQOzA4j/k0hm+IFl4jy7YM6i5vWunvImiL Rwl7FHXqJaeX2SEJMcZTi2EURPc83vP7Kj9X91ruIXT7KBzrMBLCFj/725BvHTHX2GJ7EjeN 2Z2YR01kXNf50Te40QQ878XPDlLo97fcOEUxqS8ej2SVs0dNp15SX1qr7JCEjKe4bDjg9I2b F7luIblLku2PmjBbWcx3eN2qcgB0qut83sub8q+1Hm31dxGajYeaPWo69ZIa93ysgu2+G8gu 9zz+LoU1kKo9+R6v+eonxGd4kr9lIBGT2uJxIauKPWV1esJ9er6txBUtavZcthxg1tN8Zfk9 IRGT2uIhGX+podR0Uie9P0pP3crc7w7dEcCd50P3kx1mAdjUBXHq9mDf+uqutIIJLwVFf8vg XkzKxKOqLPn3q3NEgfTEsANxYKCE7NTtEvzAZifqWkfijk7lpfxlUM3fMrgdk3LxqCZLFvVw pEX/Lth8cDuUQRvjUY9OLSJ9hw560JG+w9uhI32Ht8M/Io9+K05T5TIAAAAASUVORK5CYII= .

4) конечная линейная группа G size 12{G} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAUQCAADs AQAADwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAEMCAADrAQAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAACNAQAAAQBHUQEAAAAAAQAB AEcAjAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= , порождённая матрицей A = ( 0 − 1 1 − 1 ) ∈ GL ( 2 , k ) size 12{A= left ( matrix { 0 {} # - 1 {} ## 1 {} # - 1{} } right ) in ital "GL" left (2,k right )} {} VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAd4RAABl BAAAfwAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAN0RAABkBAAAlQAB AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNl cmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAAB AIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAIQAAACVAgAAAQBBMQEAAAAAAQAB AEEAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACm AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/ ////AIYAAQAEAAAAAAAAAHIAAgAaAAAA3AEAAJUCAAABAAAAPQBPAQAAAAABAAEAPQCMAAEA AAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAArwEAANcDAAD//wAA AAAFAAAAAAAAAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgAB AAQAAAAAAAAAcgACABoAAAAaAwAAaQMAAAEAAAAoAJIAAAAAAAEAAQAoAIwAAQAAAAAAiwAB AAIAAAAfAIoAAQA+AAAAAwA4AAAADABEZWphVnUgU2VyaWYAAAAAAACmAQAAAAAAAAAABQAA AAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAA AAAAAHIAAgAXAAAAuQMAAMIBAAABADAMAQAAAAABAAEAMACMAAEAAAAAAIsAAQACAAAAHwCK AAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKYBAAD//wAAAAAFAAAAAAAAAP8DAAAA AAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoA AABOBgAAwgEAAAEAAAASIk8BAAAAAAEAAQASIowAAQAAAAAAiwABAAIAAAAfAIoAAQA+AAAA AwA4AAAADABEZWphVnUgU2VyaWYAAAAAAACmAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAAB AP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAApgcA AMIBAAABADEMAQAAAAABAAEAMQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwA RGVqYVZ1IFNlcmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAAA/wMAAAAAAAAAAQD/AwAAAAAA iAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAALkDAADTAwAAAQAx DAEAAAAAAQABADEAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1i b2wAAAAAAACmAQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEA hwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAATgYAANMDAAABAAAAEiJPAQAAAAAB AAEAEiKMAAEAAAAAAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVqYVZ1IFNlcmlmAAAA AAAApgEAAAAAAAAAAAUAAAAAAAAA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAABAIcAAQAF AAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAKYHAADTAwAAAQAxDAEAAAAAAQABADEAjAAB AAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAK8BAADXAwAA//8A AAAABQAAAAAAAAD/AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYA AQAEAAAAAAAAAHIAAgAaAAAA/ggAAGkDAAABAAAAKQCSAAAAAAABAAEAKQCMAAEAAAAAAIsA AQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKYBAAD//wAAAAAFAAAA AAAAAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAA AAAAcgACABoAAACdCQAAlQIAAAEAAAAIIi0BAAAAAAEAAQAIIowAAQAAAAAAiwABAAIAAAAf AIoAAQA+AAAAAwA4AAAADABEZWphVnUgU2VyaWYAAAAAAACmAQAAAAAAAAAABQAAAAAAAgD/ AwAAAAAAAAABAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA AgAaAAAAwAoAAJUCAAACAEdMaQIAAAAAAgACAEcATACMAAEAAAAAAIsAAQACAAAAHwCKAAEA PAAAAAMANgAAAAoAT3BlblN5bWJvbAAA3AAAAMkBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAA AAEA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABw DQAAlQIAAAEAAAAoAEsAAAAAAAEAAQAoAIwAAQAAAAAAiwABAAIAAAAfAIoAAQA+AAAAAwA4 AAAADABEZWphVnUgU2VyaWYAAAAAAACmAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAABAP8D AAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAA2Q0AAJUC AAABADIMAQAAAAABAAEAMgCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPgAAAAMAOAAAAAwARGVq YVZ1IFNlcmlmAAAAAAAApgEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAQD/AwAAAAAAiAAB AAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAABcPAACVAgAAAQAshgAA AAAAAQABACwAjAABAAAAAACLAAEAAgAAAB8AigABAD4AAAADADgAAAAMAERlamFWdSBTZXJp ZgAAAAAAAKYBAAAAAAAAAAAFAAAAAAACAP8DAAAAAAAAAAEA/wMAAAAAAIgAAQACAAAAAQCH AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAADQDwAAlQIAAAEAawABAAAAAAEAAQBr AIwAAQAAAAAAiwABAAIAAAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAADcAAAAyQEA AP//AAAAAAUAAAAAAAAA/wMAAAAAAAAAAQD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA//// /wCGAAEABAAAAAAAAAByAAIAGgAAAA4RAACVAgAAAQAAACkASwAAAAAAAQABACkAjAABAAAA AACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA= .

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5)циклическая матричная группа 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAOwADAP7/CQAGAAAAAAAAAAAAAAABAAAACAAAAAAA AAAAEAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// ///////////////////////////////////9/////v////7///8EAAAABQAAAP7///////// /////wEAAAD///////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////// 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Спасибо

за внимание!