Централизатор. Пусть
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
– непустое подмножество группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
. Совокупность всех элементов группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, перестановочных с каждым элементом множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
, называется
централизатором множества
в группе ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
и обозначается через
![](data:image/png;base64,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)
.
Лемма
1. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
– подмножество группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, то централизатор
![](data:image/png;base64,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)
является подгруппой.
2. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAYAAABPwleqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADBSURBVDjLY2hsbGQgFzMMI80dHWk8HjIMexgYGP6DsHF0g099faykm1teGEHNUcYMCxmMoxai8KGG4NUMssGIQfaUR169IbJL3OWMZ8XW10vi1Qx3MpLNIBzt4ZGc1tHBQ9DZDdHGPmD/ohlAdGjX53kYyjIwvGGQ8diDbiNRUUXIAAznoodojptsMQOD7BvkAMSqGaQQXTPE/8Z30UMaQzMkPhEKcYU8hmZQ/BobR9eDaQaGu7DUhS/Eh1OuIhYDAA5loIZkZtdjAAAAAElFTkSuQmCC)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
– подмножество группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACsAAAAUCAYAAAAUccS4AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFHSURBVEjH7Za9DcIwEIVvgCyAhNkhXE9pXKRGChEViAqlyQAmXWagYw8mYAMKNmAHwIkMxomNSSwkRIpX5O/8fPfdOZDnOfyKfsZob/brZotiHbAhHADgKoTJNuJ8MaA0nfk2MEfYy3UahfO91WwZQHlJBhSmfRoVSZmOcLfgfCCuN5RkMGSHdVEE8lpdsxZAZHAM5MhSHpqC+hJPWchiPlGrSegmezyP41Bd04yAklmhhLGl3HGrcgK5qAloMk6AnGzvNN7cJhjpvNhKqWekjUoEAM+26oF9p3BRGTJtrKtRUzU/Gl0uhlOKK1vpnNi99wkCnN81cC1L+gdVecy8+TBbYWdHoGZWHxUugXxgoI9KJ7NVRz+NubDUtcEkai4z/JUbTLjkp+kE6XQSaSjpp2SpN838Xz8yj4lhzWqlrmz3v4i92btu3t3mlbzwuisAAAAASUVORK5CYII=)
, то
![](data:image/png;base64,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)
3. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
– подмножество группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAATCAYAAADiQ08DAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEtSURBVEjH7VVBDoIwENwH+AET6h9IeUPtgRdgvBlOhAtnU7n1Dd646R/8Cj/wD2pLWqR2AeMBTDhsSLpLOzsz20JZljDnmDW4BeBfA0woVADwsEGTSsp0xSg7plKuJgN42tFYAwr47R2IWScsKyZj0ICgu1Psy2eMFG7upwNFzkMCcO9IBbTeC7FGa19SYvvljB54LkIUoKUfyN0U9krirA17zg/+6yFRVKtOzdfN+8zcF6qeB3BzG7LrPQr0Suca1ubFfk0B6u7GuMS2HpEXGxAUoOkMM7M+cLM9j2VwCKBWCpEfNWsUkQu2oW6AkKtr6GEPtt7+kBk5y0u3olrLTPhVseSbrkaW8aZvbdH9x9gJU8uhubnRXQNjXmynfviaQV8Qzw2xvMULwDnFE2GxU/M1D2Z8AAAAAElFTkSuQmCC)
, то
![](data:image/png;base64,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)
Центр группы. Центром группы
называется совокупность всех элементов из ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, перестановочных с каждым элементом группы. Центр обозначается через
![](data:image/png;base64,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)
. Ясно, что
![](data:image/png;base64,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)
, т.е. центр группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
совпадает с централизатором подмножества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
в группе
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
. Кроме того,
![](data:image/png;base64,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)
.
Зафиксируем в группе
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
элемент
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACbSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xkYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6sJ5hdQYBjHxqZERtYb4nEeKMRA2PhuZH2kIUQjqpOHaooghAFL3Brb++O5vwAAAABJRU5ErkJggg==)
. Пересечение всех подгрупп группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, содержащих элемент
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACbSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xkYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6sJ5hdQYBjHxqZERtYb4nEeKMRA2PhuZH2kIUQjqpOHaooghAFL3Brb++O5vwAAAABJRU5ErkJggg==)
, назовем
циклической подгруппой, порожденной элементом ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACgSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xgYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6kJGACMML8A2f+NY2NTIiPrDfE4j+E/JNSM70TWRxpCNBrfwem8YZSMAIJuGVXT6DGkAAAAAElFTkSuQmCC)
, и обозначим через
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAAAkCAYAAAD8fqYDAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAKnSURBVGje7Zo7csIwEIZ1AC7ADEqXA2DV6TKOmeECwFCFocq4Sc04dBQ5QbqUnCGcgBuk4AbcgYysEZYXrSxhHpajYhsjY+nb3d/alclyuSTBrmMBQoAb4N7dVqt5J+mRDSHkUDI2/g5wa9iYkW8OksZv7y6/BbhWYNlummVdbEyWJn1KyP6WgI0TZpOPoc2ffEzY8F6plz+b0H2SZn27seRguy4Y+dKkg7DrRrjnwHJxxsU11mGuOZAzAuEtpu86J2LXtXDzCVO6tomEk7SjyXq+WnVuBVemuotTRfSaJcTFKSZn6T10ZorzB91S01wkoc49WTbtMkJ20InYdS1cPjgi0Q/mVZECXF/oPo7ZgvSSjRqppvuhPl1iu3QWqDza6a+7Q06jvSoLrCIPapv0mG5snchvKlxjYBjWaqWZUFckbF06iOilW1fNbipcuVYYSFWSUIKbTxSkORb6YoLRVpcO2EN9lQVsvM2zK2UBi1oMRttkIV+PJuiw604vNBWujMp8Iz6dzkaj8gSrXoi+wcXeLaZ3TuVWTL1JVjXC2G6UjfoC8Olb0oetmC3cYmdUli143bRer4uIW+0WLtpb8KX89RKuL40bb+H6YAFugBvgBrgBbrB/DTevsB5evq69J28VtKJ64pVk+ng8Zld6ANhhplqJ8qprkiSvdeG3Ci6sLtXunelISDilAA5bAK2FezwSHww+q5pCsARXixus73HNZlPjtVEuXG1llppJSsqXm0zC1EjVndSqDpCOlNJSF3ij4aYxm0kQHAKPvJNPloCeSphYAwr2o2Hpjh0atA7uhLGs0EuxN+VwYhYv4OJhett26eCpyiXbpo2GezwaotE2omQrIwrKQpo+P8l05pFb/C4kQPuBHionbkVJ2OeGIiLADeZgf21f/cAbrF7EAAAAAElFTkSuQmCC)
.
Теорема. Циклическая подгрупппа
, порожденная элементом
, состоит из всевозможных целых степеней элемента
, т.е.
Следствие. Циклическая подгруппа абелева.
Порядок элемента. Пусть
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACbSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xkYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6sJ5hdQYBjHxqZERtYb4nEeKMRA2PhuZH2kIUQjqpOHaooghAFL3Brb++O5vwAAAABJRU5ErkJggg==)
– элемент группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
. Если все степени элемента
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACgSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xgYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6kJGACMML8A2f+NY2NTIiPrDfE4j+E/JNSM70TWRxpCNBrfwem8YZSMAIJuGVXT6DGkAAAAAElFTkSuQmCC)
различны, т.е.
![](data:image/png;base64,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)
для всех целых
![](data:image/png;base64,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)
, то говорят, что элемента
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACbSURBVDjLY2hsbGQgFTMMQU05brLFDAwM/xkYZN+4uRnXMsh47Enr6ODBqqmjI43HQ4ZhD4Nx1EIQv74+VtKYgeGurFtOMU6boowZFsI0IBtiHN3gg1VTQ7SxDwOD8d3Y+npJmFh9noehLIPRKWQxFE24bEEWw6sJ5hdQYBjHxqZERtYb4nEeKMRA2PhuZH2kIUQjqpOHaooghAFL3Brb++O5vwAAAABJRU5ErkJggg==)
имеет
бесконечный порядок.
Нормализатор. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
– непустое подмножество группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADMSURBVDjLY2hsbGSgFDOMEEOAgDHKmGEhkP4Pxf8YjKMWlpen8boZu9WmdXTw4DWkIdrYB6xRxmMPsmKYuKxbTjFel8AUGkc3+GCzIMdNthhdDkVBfZ6HoSwDwxuQs3F5M8/NOMUjr94QpyGQMDC+G1tfL0lWwHZ0pPF4yDDsQQ8HuDgigIHY+A6yRQiv1MdKGjMw3MXlFVyBSpIhoADF5VUsYSL7Bj3g4F7CYQFq7MBcg2QjKNHBYg1XtGMNbbSUCkmtaAE+EnMxKRgAd6jQkCN0LT8AAAAASUVORK5CYII=)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC0AAAAVCAYAAADSM2daAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFmSURBVEjH7VY9bsIwFH7scAAk3Dsg74yuBy5QEFuVCWVhYEImW87QjbF34CjlBpwhFMc4OM7zS4JaASLDUxT/5XvfjxVIkgSerZ4OcAf6po8D9GYcdufn6VIZ8NluvY4GgotNlKb9hwK9nfNpDnQk9y44O87EcvVQTFtgfL6dYvNLwVahuT8DrWI5ZgDHkszAfxZKDYNrzzYInRcL/iljNW4FupDOK0yyfK0nMVXGw/yANXRzEA1gdrSd+u9upWnUpwKDrZcj2PtNFuMlknClKqDtZpdRAxpnRqnFkAMcEFVQexTrA9ZoEsIgaH2ovo701WPfUS9rEG/vX02ZrgOtA9jEOnWhyqjA5E0y9k2FBvd01W4uYa093ZY9yj4k284eraoli7rqgqCNRIblOm8Rt01GBenKuLfHC+hV9TIplYNcoEUDDST7z1IfcuLaiZRZy5aP3wl0KLj1kt2Z5e7XtAP9CqB/AeOmW7T2S823AAAAAElFTkSuQmCC)
то
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
Элемент
![](data:image/png;base64,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)
называется
перестановочным с подмножеством ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
, если
![](data:image/png;base64,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)
. Равенство
![](data:image/png;base64,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)
означает, что для любого элемента
![](data:image/png;base64,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)
существует такой элемент
![](data:image/png;base64,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)
, что
![](data:image/png;base64,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)
. Если элемент
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACRSURBVDjLY2hsbGQgFzMMLs0N0cY+DAwM/9GxrFtOMV7NEI2ybzzy6g2x8XFq7uhI4/GQYdiDbANEs/Hd2Pp6SaI0MxhHLcTGJ+jn+jwPQ1kGhjdwv+LQiF1zfayksZz7rLSODh6SQzvHTbYYV+ji1RxlzLAQWQPcIEJ+xhWqYHFiAgxkM0biICXAhnDGIBYDAPgxcNv4hKtEAAAAAElFTkSuQmCC)
перестановочен с подмножеством
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
, то
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
. Совокупность всех элементов группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, перестановочных с подмножеством
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGQgFzMMDs1RxgwLGRgY/uPExlELsWru6EjjcZcznhVbXy8J4ue4yRYzyHjsSevo4IHxjaMbfLBqrs/zMPSIrLeDGeQhw7BH1i2nGC4fGWkIMxivn0EGyTLI3vTIqzckOcDATmYwvotuE0HNMCejBw5RmuvrYyWNGRjuogcOUZoboo19iHEyVs3guCbCyRiaIaHM8IYYJ8M1wwMJOTUhJZBBnjFIxQB8yHOGUe7KbQAAAABJRU5ErkJggg==)
, называется
нормализатором подмножества
в группе ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
и обозначается через
![](data:image/png;base64,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)
. Итак,
Лемма. Пусть
– непустое подмножество группы
,
– произвольный элемент группы
. Тогда: 1)
![](data:image/png;base64,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)
;
2)
![](data:image/png;base64,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)
;
3)
![](data:image/png;base64,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)
;
4)
![](data:image/png;base64,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)
;
5) если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACvSURBVDjLY2hsbGSgFmYYWoZFGTMsZGBg+A/Dsm45xfjECbosx022mIFB9o1HXr0hMeJ4DQO7wjhqIbHiOA2rr4+VNGZguGsc3eBDjDhewxqijX0YGIzvxtbXSxIjTlIEoGAcXsRqWEdHGo+HDMMe9Jgi5EWshtXneRjKMsjeRI8tiBexxyJOw8BRL+OxJ62jg4cYcZyGwbyCy4vo4jgNgyRGzIBGF8dn4BDL6ORiANJJ0On7HoBIAAAAAElFTkSuQmCC)
– подгруппа группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC8SURBVDjLY2hsbGSgFDOMIEOijBkWMjAw/Idj46iFHR1pPG7GbrVpHR08eA1piDb2AWuS8diDrBgmLuuWU4zXJTCFxtENPtgsyHGTLUaXQ1FQn+dhKMvA8AbkbFxezHMzTvHIqzfEaQgkDIzvxtbXS5IVsKAA85Bh2IMeDnBx5ABGswjhlfpYSWMGhru4vIIrUEkyBBSguLyKJUxk36AHHNxLOCxAjR2Ya9BshMUarmgnLqViSXgjNRcTiwEtI9O4eJHXcwAAAABJRU5ErkJggg==)
, то
![](data:image/png;base64,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)
Подгруппа
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACvSURBVDjLY2hsbGSgFmYYWoZFGTMsZGBg+A/Dsm45xfjECbosx022mIFB9o1HXr0hMeJ4DQO7wjhqIbHiOA2rr4+VNGZguGsc3eBDjDhewxqijX0YGIzvxtbXSxIjTlIEoGAcXsRqWEdHGo+HDMMe9Jgi5EWshtXneRjKMsjeRI8tiBexxyJOw8BRL+OxJ62jg4cYcZyGwbyCy4vo4jgNgyRGzIBGF8dn4BDL6ORiANJJ0On7HoBIAAAAAElFTkSuQmCC)
называется
нормальной подгруппой группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADMSURBVDjLY2hsbGSgFDOMEEOAgDHKmGEhkP4Pxf8YjKMWlpen8boZu9WmdXTw4DWkIdrYB6xRxmMPsmKYuKxbTjFel8AUGkc3+GCzIMdNthhdDkVBfZ6HoSwDwxuQs3F5M8/NOMUjr94QpyGQMDC+G1tfL0lWwHZ0pPF4yDDsQQ8HuDgigIHY+A6yRQiv1MdKGjMw3MXlFVyBSpIhoADF5VUsYSL7Bj3g4F7CYQFq7MBcg2QjKNHBYg1XtGMNbbSUCkmtaAE+EnMxKRgAd6jQkCN0LT8AAAAASUVORK5CYII=)
, если
![](data:image/png;base64,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)
для всех
![](data:image/png;base64,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)
. Запись
![](data:image/png;base64,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)
читается: »
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADJSURBVDjLY2hsbGSgFmagi2FAwAjDxIjjNCzKmGEhUPF/KP4n65ZTjE+coMty3GSLGRhk33jk1RsSI47XMLArjKMWEiuO07D6+lhJYwaGu8bRDT7EiOM1rCHa2IeBwfhubHm5FHKAw8Xr6yWJNgwpoP+h4f+4vIjVsI6ONB4PGYY96DFFyItYDavP8zCUZZC9iR5bEC9ij0WchoGjXsZjT1pHBw8x4jgNKy+PlQJ5Bd2LuMRxGgZJjEiBDQ1odHF8BjIMvVKDHAwAHJvDHhGGI9oAAAAASUVORK5CYII=)
– нормальная подгруппа группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADlSURBVDjLY2hsbGSgFDMMHUOAgBGGSTYEpCnKmGEhkP4Hx8ZRC8vL03jdjN1q0zo6ePAaUh9p7AvU9J9B1mM3suKGaGMfkLisW04xXpfAFBpHN/hgsyDHTbYYXQ7VBXkehrIMDG9AzsblzTw34xSPvHpDnIZAwsD4bmx9vSS+AEYPZDijoyONx0OGYQ+DjMce5HCAi4PCCI6N7yBbhPBKfaykMQPDXVxewRWoJBkCClCcXsUME9k36AEH9xIOC1BjB+KaO2Aby8ulYIFYneViBIo1XNGONRoxUioIoyW8IZqL6WYIAGDsxF7fD3CIAAAAAElFTkSuQmCC)
«. Равенство
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAATCAYAAAAnMdWSAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFmSURBVFjD7ZcxcsIwEEU3PTkADDoEI87gUcEJYNxlXDFuKFQKd74GXXIHLkLBDTgDBNmR8SjSSnI2lV3sMHyP/fdJf4WBqqpgrDVa8Al+8I0Ab6ZidLKGCX0HNbDlcHqaPH7qzrL9AdOpitp3cCP7jB0A2E2UahWjUxWl7992gW9PsTrp7hP5emfK1vrflcrnHODKd8dNjI7NrV3YfbG+MQy/4I87vtEz049Ppy3Fuajr2Uvj11zKRb/xTldqHpjbu7eQ3YvxjWXwxr6Zn2cT5hM5eOzmH/8e+UjfEIMXXpVixQBurlOzrouZWMLZvhYb+aGxT/XFGFB4Y+QCaR/KLvap2kYrfNoOjX2qL8aAwpcZ/1iv2aerkSZG1uxgOulPXIIvxuCEN4eHjkqz0kx8FVK+6wfplZUyX+iI2VHy6VSV4hticMK3Lwmv2JnYmDem7roVT1unXoAU3xDD9Mdmgp/gxw3/DWY6itpG6XCSAAAAAElFTkSuQmCC)
означает, что для любого элемента
![](data:image/png;base64,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)
существует элемент
![](data:image/png;base64,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)
такой, что
![](data:image/png;base64,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)
.
Теорема. Для подгруппы
группы
следующие утверждения эквивалентны: 1)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADJSURBVDjLY2hsbGSgFmagi2FAwAjDxIjjNCzKmGEhUPF/KP4n65ZTjE+coMty3GSLGRhk33jk1RsSI47XMLArjKMWEiuO07D6+lhJYwaGu8bRDT7EiOM1rCHa2IeBwfhubHm5FHKAw8Xr6yWJNgwpoP+h4f+4vIjVsI6ONB4PGYY96DFFyItYDavP8zCUZZC9iR5bEC9ij0WchoGjXsZjT1pHBw8x4jgNKy+PlQJ5Bd2LuMRxGgZJjEiBDQ1odHF8BjIMvVKDHAwAHJvDHhGGI9oAAAAASUVORK5CYII=)
– нормальная подгруппа;