Подгруппа
![](data:image/png;base64,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)
полунормальна, для неё супердобавлением является подгруппа
![](data:image/png;base64,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)
. Так
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
.
Итак, в нильпотентных группах подгруппы, обладающие супердобавлениями, могут быть ненормальными.
3. Факторизации групп дисперсивными и сверхразрешимыми подгруппами
3.1 Силовские множества и их свойства
Определение 3.1.1 Множество
![](data:image/png;base64,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)
, состоящее из попарно перестановочных силовских
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAARCAYAAADUryzEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADxSURBVDjLY2hsbGSgBDPQxAAgYIJixlUINhNRBtTneRjKMjC8Amr4ZxxZ7xttzLAAyP7LyGB8N7a8XIooF9TXx0oaMciekpNjuG4c2+Dd0ZHG4yHDsEfWLaeYOAMijX0ZZI1PRubVq4L4cAM88oqIMiDKmGGhcXSDD4xfneViJMcgd8Mjr96QcBiAnW+0O7a+XhLEBwVihBHDIgbj6AVEBSLINllZj1WhaWl8oJCPNGZayMBgfBtmIEEDctxkS0ChDsfGUfOJTgfgwJKVXYnNrwQNAPkV6HxjmPNJNgAY8vPhzpZ135XW0cEzcHlhaBkAAHQ7gLaXyiXAAAAAAElFTkSuQmCC)
–подгрупп из
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEzSURBVDjL1ZNBasJAFIb/xK3QtZC5Q5tHbyBxQOkuCxOyKmQhIYumO5Eku9zC2qVHEHsRiSeoR6i1byTVNGTqolBw8XgwzHx5///+IM9z/LVwHZAlYALonGtpHh8BxkWIuqQejclccP84FfkvYejeOOTMwqLoaiEKMBvfPXDfQwxW9ctZQEOm74UTJdpJzgDjk4Js2DZl5IiE/GykhUwnfVvA2IG8uc6nSN4/yiS71UI8AntA2yBNe78b7HZaIUURdqWFNSy5rvugzgcCq6NHwIGlcrc39Q+dIGka9AjYspRFm4zMpxEDDkLGT9qcVJAStvfazEFl6DNLLZtS2zyZGxC7/mRK9fNK6pvKycXEqmlsYGPALt04tr6N5K0Rb+1dt/bWNaqJfiRVVSN4V/gX/yvkC7qHujwxDs7nAAAAAElFTkSuQmCC)
, в точности по одной подгруппе для каждого
![](data:image/png;base64,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)
, вместе с самой группой
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEzSURBVDjL1ZNBasJAFIb/xK3QtZC5Q5tHbyBxQOkuCxOyKmQhIYumO5Eku9zC2qVHEHsRiSeoR6i1byTVNGTqolBw8XgwzHx5///+IM9z/LVwHZAlYALonGtpHh8BxkWIuqQejclccP84FfkvYejeOOTMwqLoaiEKMBvfPXDfQwxW9ctZQEOm74UTJdpJzgDjk4Js2DZl5IiE/GykhUwnfVvA2IG8uc6nSN4/yiS71UI8AntA2yBNe78b7HZaIUURdqWFNSy5rvugzgcCq6NHwIGlcrc39Q+dIGka9AjYspRFm4zMpxEDDkLGT9qcVJAStvfazEFl6DNLLZtS2zyZGxC7/mRK9fNK6pvKycXEqmlsYGPALt04tr6N5K0Rb+1dt/bWNaqJfiRVVSN4V/gX/yvkC7qHujwxDs7nAAAAAElFTkSuQmCC)
, называется
силовской системой группы
![](data:image/png;base64,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)
.
В своей книге Дерк и Хоукс использовали название «силовский базис» вместо силовской системы
![](data:image/png;base64,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)
. Введем следующее определение.
Определение 3.1.2 Силовским множеством группы назовем множество силовских подгрупп, взятых по одной для каждого простого делителя порядка группы, вместе с единичной подгруппой.
Таким образом, если
![](data:image/png;base64,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)
– группа порядка
![](data:image/png;base64,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)
, то множество
![](data:image/png;base64,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)
будет силовским множеством. Здесь E – единичная подгруппа группы
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
– силовская
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAYCAYAAAAYl8YPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADlSURBVEjHY2hsbGSgFmYYNWzUsIEyrD7Pw1CWgeENAwPDf+PoBp8oY4aFIDYDg/Hd2Pp6SZJdVl8fK2nEIHtKVpbhJsjAjo40Hg8Zhj2ybjnFJBvWEG3swyBrdCoyr14VxKfIMJDXQC5C9brsTY+8ekPSwgzsRaPdyOEDDjfjqIUwPsilbsZutWkdHTyEI0DWYyVMISQC8Ac+TsNy3GSLIbEHxUguQnGpjMcevC4DB7Ss7Ep8YQOLIOQwxWoYuhdx4Whj43p0b2M6HeY1NC+gR5CxcXQ9VbITOA0CLURPc6NFEOkYAMZ2oHtcn+qwAAAAAElFTkSuQmCC)
–подгруппа группы
![](data:image/png;base64,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)
и все числа
![](data:image/png;base64,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)
различны.
Из теоремы Силова следует, что каждая группа
![](data:image/png;base64,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)
обладает силовским множеством
![](data:image/png;base64,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)
. Если дополнительно
![](data:image/png;base64,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)
для всех подгрупп из
![](data:image/png;base64,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)
, то силовское множество превращается в силовскую систему, см.. Известно, что любая разрешимая группа обладает силовской системой, и наоборот, если в группе имеется силовская система, то группа разрешима. Кроме того, если
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
– силовские системы разрешимой группы
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEzSURBVDjL1ZNBasJAFIb/xK3QtZC5Q5tHbyBxQOkuCxOyKmQhIYumO5Eku9zC2qVHEHsRiSeoR6i1byTVNGTqolBw8XgwzHx5///+IM9z/LVwHZAlYALonGtpHh8BxkWIuqQejclccP84FfkvYejeOOTMwqLoaiEKMBvfPXDfQwxW9ctZQEOm74UTJdpJzgDjk4Js2DZl5IiE/GykhUwnfVvA2IG8uc6nSN4/yiS71UI8AntA2yBNe78b7HZaIUURdqWFNSy5rvugzgcCq6NHwIGlcrc39Q+dIGka9AjYspRFm4zMpxEDDkLGT9qcVJAStvfazEFl6DNLLZtS2zyZGxC7/mRK9fNK6pvKycXEqmlsYGPALt04tr6N5K0Rb+1dt/bWNaqJfiRVVSN4V/gX/yvkC7qHujwxDs7nAAAAAElFTkSuQmCC)
, то
![](data:image/png;base64,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)
для некоторого
![](data:image/png;base64,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)
.
Пусть
![](data:image/png;base64,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)
– некоторое множество подгрупп группы
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
– нормальная подгруппа группы
![](data:image/png;base64,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)
. Воспользуемся следующими обозначениями:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIAAAAAYCAYAAAAyC/XlAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAARySURBVGje7VnJTttAGP4NHCmcaTJ5BpLh3h6QM4ilJ6t1otyQVUXIEktlIUQdbjwC5cJyacUbQOEB6ANUKssDAH0DKHRmHMdLvIwThybFSHNgnORfvu9fDVtbW5Cdl3syJ2QEyJyQEaDLowAMAz9HQ4Ng9KDp29cEMDRlrIzghDr0AclLq/1u8KDp+08IQP8kaEWJ+zxJHmca2qsyQkfTmjHFnquyvKhtb4+mreQRwBCAMhz+rF3H9nuQRPWNywxJZKYGlAAmjvwnqWMCsB8xtOkpBNIdixL7UOl/iur6ghsIU8Xz7ueAq/u9AF/FQweAyidBYC0RtMxlU6uZHrhmzrL7OkErEr+T6LPCDVltTIroa5q1CYyrZpROMTIf3TLT8kEoJu+NdzYmTC92b+EkVt48/zBvfSjBITcAV/Y8jqnhWW50Xj5TNGOM3VUxNmumOWGTAdcas2kToML0QTMnUZmlwXVDd9P1jZKTnZQcBrhAZGnZvhPRV4QASWR2ewQweXBj0lglkwjgjpa3tcQE2FSLlDnSEyL1lXCjKcNLlUOeTgt4V9H1vK4r+akC2Uk7/XMwAK6xas5HfW57WxslCL7bRrM6TwpTX2yw7c+I6CtKgBCZO26Z4qk8uLw5WVZ6FMHELjkVDAeQJ6cieHgNysMpAL4OM8I2GgD9JrpZpIL2rHRUuogzPKA++k57yrJ1EmGzTtMvi4SaUcvNIPQ1KP2K6CtKAFGZ/mjeqBNcALhxp/IwfTrBpPUdwXLcFm1xzFmS0SqrfbjamEuUylvODzm+9GZHCy8BuLonlpKZj9FtN7U3CQFa0Scok0dzSC8TlQGTYNL8zmXiEtBrAnRS+6xoKfwUcS6vv1J8uUiTAE2ZVyIyWWTKWP6saNq4aAloYnKVFBPWG9CsccHKXdxE4FFwhs3HeXKmGcarCGFrlPF3SaMsaQngRtBoYQ4TAgLwMcZUf1oLn4MASWVaYEpXwRkwvAR0gok18loNqT2hCDWBVleJbmlni0OXKHnpNChd96IEWB1t8VwxzFwcEMyBvCZH1Mu0CNCJTP67hfJuFJCB3+sAk436NEaAz0X84O9OpVb6MIyce5HB1qd0Ht9ngqzPii8buhoDWUeLKwdBJYJ18zYQDmHQL/doliYBupFpRTP6FgZkxMQgeVN6NCauxlloy+kYrpMiUTffsnTMjCwC+sG6Snc3ai0YYJgtJbCsL/Ya/LCOlhm+zkfW0nXNaLx2RcO4nIczJH/81OkmLowAaci0ohlfNoGM7wEcTEZCMeGLIBixMQmbnJzx0yvLk26944mzyWqOGsfulJ3msiOuC0ZEX3HflWht8+vhv+90ExdEAAGZ96IyrWi2Px/dAwhicu/XKyhrOr+FL92y+vpFhd0FF9XNhWeVKTgF9ONpbQ4D9gAsi5nq3Bs3SfveIGvvjq+iGsGMAM6kZZUp5/1EWxbzNdsDYRh/yYHIcS/eNP5PBLBeBiUrfQOZ5p6BAJsvxd4M9Bd+MidkBMic8JLPX6RF8Q3QLHSZAAAAAElFTkSuQmCC)
где
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAATCAYAAABLN4eXAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACpSURBVDjLY2hsbGQgFTMMUU0dHWk8HjIMexgYGP4zMMi+8cirN8SrqT7Pw1CWQfYmTGGOm2wxg3HUQpya6utjJY0YZE8hm9wQbezDIOOxJ62jgwerJnRTCWrCZgsIRxkzLJR1yynG6jywLWCPY2Lj6AYfDE2w0EI2EZvTUDSBnGbMwHAX2URcBpEUajidB7MJrIHB+G5sfb0k3hQBUQj1PFqwD4dUjg8DAJCAZV8o8UMtAAAAAElFTkSuQmCC)
– некоторый гомоморфизм группы
![](data:image/png;base64,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)
в некоторую группу
![](data:image/png;base64,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)
.
В разделе 3.1 изучаются свойства силовских множеств, которые необходимы при доказательстве. Для формулировок теорем потребуется следующее
Определение 3.1.3 Пусть
![](data:image/png;base64,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)
– некоторое множество подгрупп группы
![](data:image/png;base64,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)
. Подгруппа
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVDjLY2hsbGSgFmaguWGrGBiYGBgYmBH4PyM+cbyGRRkzzGdkYPgL1PAfiP/KemQV4RMn6M0cN9kSBgbZ1x7FDQbEiOM1LMqIYRGDcfQCYsVxGlZeHitlxMB41zi2wZsYcbyG1Uca+zIwGN0NzcuTQQ7w2khDPwYG4zux9fWSRBsGDOiFDAyMoID+g4wZQQGPw4tYDevoSOPxkGHYI+uRV4TpRYa7xtENPkQbVp3lYiTLIHvTI6/eEFm8IdrYBxyLaOJ4DQNHvazH7rSODh5ixHEaVl8fK2zMwLBM3i0vjBhx+ufN4WcYAHmDw8jnpPAjAAAAAElFTkSuQmCC)
группы
![](data:image/png;base64,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)
называется
–квазинормальной, если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAARCAYAAACPZ7H1AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAJMSURBVFjD7Vcxb9NAGH2XIqYWZmjObMyNLxIScxSdlFZMFmoqM2awIldpizx0sLNlzcZUoAuSfwGi8DcQoIYZ6E+gUO5sGRL7bJ9LAouHtzyd/b73/H2fbYzHY9RQow6hajgWsAbgxh+EDSUfho0QaCyevSLybJYHWXbxYZijneGzdUd+LDVfGE7fxHMC/BAXXAlc0u5wpOT5cDTs0pHgLhOO2X5PnnU4PRD8T5GJOG984YfB1rLDGfISbRJrGwbeE7JYd+SnjRMVXzpWwvQRCL1Im1Lxvs16BPSi4xybCed51qYJfEqCXRWCIu05s1X8lIYjuuQUbO+FDj+ZDNY5xZkI4igqbmDd5kb7me37d1a9FxTat1TaVfwUhuN59l0TZMbsoKfDS7iildHsvrM9e5NT+kpnjLK7LQXLWtMJqEz7On5yw/F32Q5gzrZdl84X6z1uPQJh56qOiNtb7hjjq+5+kTtM3Pd7Ltr9E/3Rktr0m0o79tP6XMVP/kJmeBkv0sViowJyWtB1rSYjOGe7/s6/fuWWaUd+SDU/ynA8b7DBm3hLuXuQbUHM2F6wrSwO7DVjeINW/1TX1DLGakHbzGpfx09uOMdOxzRgfEi3Z9ya2a2eFCdbM5p9sJnuIv7bsdLRln4o6EddP4XhOB36FJSfDSaT9SJefBgQ17V/FxcJurwlg51/ra4CKu3gkG/JENLaTjfHTw6fG45sQQbs0477sIz37Qf3xZPaH3jeRsJNp9Ob/B6epK9fNnS1q/ip/63qH886nDqc/41ft3x7AMV3VLUAAAAASUVORK5CYII=)
для всех
![](data:image/png;base64,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)
. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAARCAYAAAA7bUf6AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEjSURBVDjLY2hsbGSgFDPQ1BAgYMKCGYkyZBVQcXWWi7EsA+NroKa/yNgwstaPgWEVE15DQDZFGDEsAtL/GIyj5iPLNcQae4MNk3Hbm1ZezovTkPpIY1+gwv+yHnlF2JwNMogRZIFR1CKshnR0pPF4yDDsYWAwvhtbXy+JzRCwGlmG3QwMsm888uoNMQypr4+VNGZguMsg47EnraODB1eA57jJFoNcaxzd4EMbQ+BOJWxICdA7r7F6BxEDsq9csqqNsRlQXp7G6y7DuAc95jCiGGKQ8e3Q8nJp9PQTbcywAGYAcsJDhEmeh6FHZK0DKGWWl4dKGzHInoI5GRZzkMQGSYzGbnkpWA2RZWB4hUihsq+QDXGXZdiFnHqR0xLD4M/FpGAAOCiKEiedWX4AAAAASUVORK5CYII=)
– множество всех подгрупп группы
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
–квазинормальную подгруппу называют
квазинормальной.Лемма 3.1.4. Пусть
– силовская
–подгруппа группы
и
. Тогда
– силовская
–подгруппа группы
, а
– силовская
–подгруппа факторгруппы
. Лемма 3.1.5 Пусть
– нормальная подгруппа группы
. Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAQCAYAAADJViUEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAEDSURBVDjLY2hsbGQgF8MZ/xkYGBkYGJhx41VMODXXxxp7A3X/BSoEmsPwBwMbRy3EqRmEszxkixgZZF+7ZFUbI4s3FHsYyBpH9eLV3NGRxuMuy7CLUcZ9T2haOR+yuIepR3VaRwcPTs1g59fHShozMNxmMIpYBPQBI1EBhmIA1P+GkbV+QP8ykqQZhHPcZEsYGGRfeRQ3GJCkuTwtlM9dlnG3cWy9N0nOXsXAwBRpzLRQ1iOnEEUhFuejcEAJJctNvgQYp/PRFVaHhgIdUi+JVTPI5NpIQz8GGbe9oWlp/MipKy8vVMaQwWg3Ts2ghCDHwPASa+oCYVmPnQTjmayMQXfNACowQu7MEHTCAAAAAElFTkSuQmCC)
– силовское множество группы
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
является силовским множеством факторгруппы
![](data:image/png;base64,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)
.
Если
![](data:image/png;base64,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)
– силовское множество группы
![](data:image/png;base64,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)
, то
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC0AAAATCAYAAAAEaoRHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAIMSURBVEjH7VbNTsJAEB44C3eV7TvQ0bMXs6wRvNXYNj2pPRjSg5D0gFi4cfEB5CJw5BEQ30OjvoDxFfzZRWqWul044AEjySTQbme+me/7pkC73YZVi5UDvPqg+SfDI5sew+zSi2tyDzW1v7+0PNznB955fPB4+xHo9JcJOIq8dQR4nOQusLHf6azF9zodf61EYDTFI2rfpMojYOQ8A+R196xhzhQIWJGgc/UbVLdcLIOipggHyRULoqJW06LDPdFhoTT2wzAnX2fILuRpJGhWSSuzCOiA4olhwD2YziDJxJbBrlU102h7AvNosEhhob3IxkpSTmhHlXk+mAyDucdf08ZnLww3ZHaRBicLb49Y30W7eTDHSBkHoQ+kNFJqkvtA13gMLB4UodW6zIBKGtqVV6WkzrX2kvbg9EyNsOA8lXruEQ6kppNGnN9F6MWGFE1TpM1UOaouhqGfYwRu0Wvta91vlLppib8nbmDXi6J11T0Z2MTsQB6EIXXSUIIWdNqY7esmGLteN0WZDXRbZVXTlAaHM/rmgxKGrFI81TH8A/BEFuj2kgcblmXKE0sDs2hzkc12ksDi9WegnsGZH027eAAFemf5fl5eX2FobZpg3i4DtBiM71t5Lo1LUUe1ueaxPPsCAXhRvg1FJDaEMJlO8/ImkkFMN9M0Lz4m9R6w7WOdNP7/5f2D/ougPwFUdbPsKIm6egAAAABJRU5ErkJggg==)
является силовским множеством подгруппы
![](data:image/png;base64,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)
.
Если факторгруппа
![](data:image/png;base64,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)
имеет силовское множество
![](data:image/png;base64,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)
, то найдется в группе
![](data:image/png;base64,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)
такое силовское множество
![](data:image/png;base64,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)
, что
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEQAAAAXCAYAAACyCenrAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAJWSURBVFjD7VfLTuMwFHVZA3ug7j80d9gyG+QagdiFmSTKCiYLVHmBkLKA4nbXDR8ws+Gx5BN4/MeMGH4A8Qs87NAIkwbbacMKV7pS5frkuOfec6+DBoMBcvEWTgQnSAVBxGdGF19RkEddOMu4HuJCtUtD30MuaukhVXl0/BcFnHz2pNgxQfoxrIsfn0Q8l/YQCM/qEKQKD+fxAiB0m6036XUyHM7mvw2HyWwHo8vRswQuOo0AeMz5QnVseFJqGUbxXgPhh9XdA09d54y2MYTHdZVlVZ5+BBuoZL+MEPAxZbydCQARnwT7YQ+Ryq1J5Zqd6yRN59R1CvRQVXmaqMrDCOy0Wugv8sLzdwIKEb616G+5X/5xiPobY+JbYLVNdVRq/5H387zoSb0Pq/UdW55MJBptv2Ya7uI0XVQrCgjbkd9Vu1TFGqdM7vN20NvUCRICOtHeXRRvTsqTHzwXEJPuvpp9nV1ssNZjt0vwvvDefRmozjDxqAePAJ3mDVJmnwDpmexiwloJkqbJHMXoCuL++mdZxoanePCs6SL8TzZIG7vYYI2CSD8HMHOGKdszZXcay9jwyFInhG296wlCQNkguwR+KXY5mgRr8y7TyEpYzPTixgPf94pZmOZyZsPDA/q9ePB8jLag80dmX2abBnxlTBALrFGQXtDeRE1y4yfJvFr6aeoveci7qksQI4+YBkniz4uSP5J7yqZTXlmMkC31XFJsW6xWkFePofsPyx93Luu4h9jwsGj5x9sa3BYTwejytsx+2b1lNLmMWPdy5952nSBOkM+IF5FDg65VAoLRAAAAAElFTkSuQmCC)
.
Если нормальная подгруппа
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAARCAYAAAA/mJfHAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADnSURBVDjLY2hsbGSgFmaguWGrGBiYGBgYmBH4PyM+cbyGRRkzzGdkYPgL1PAfiP/KemQV4RMn6M0cN9kSBgbZ1x7FDQbEiOM1LMqIYRGDcfQCYsVxGlZeHitlxMB41zi2wZsYcbyG1Uca+zIwGN0NzcuTQQ7w2khDPwYG4zux9fWSRBsGDOiFDAyMoID+g4wZQQGPw4tYDSsvT+P1kGHYI+uRV4TpRYa7xtENPkQbVp3lYiTLIHvTI6/eENPrwFhEE8drGDjqZT12p3V08BAjjtOw+vpYYWMGhmXybnlhxIjTP28OP8MAKHXDpVY4CS4AAAAASUVORK5CYII=)
группы
![](data:image/png;base64,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)
имеет силовское множество
![](data:image/png;base64,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)
, то найдется в группе
![](data:image/png;base64,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)
такое силовское множество
![](data:image/png;base64,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)
, что
![](data:image/png;base64,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)
.
Если
![](data:image/png;base64,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)
– силовское множество группы
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
– некоторый гомоморфизм группы
![](data:image/png;base64,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)
в группу
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
является силовским множеством группы
![](data:image/png;base64,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)
.