Знакопеременная группа

Знакопеременная группа перестановок (подстановок) степени n — подгруппа симметрической группы степени , содержащая только чётные перестановки[1].

Обычно обозначается .

Свойства

  • Индекс подгруппы знакопеременной группы в симметрической равен 2:
  • Знакопеременная группа является нормальной подгруппой симметрической группы (следует из предыдущего утверждения).
  • Порядок знакопеременной группы равен:
  • Знакопеременная группа является коммутантом симметрической группы:
  • При знакопеременная группа является простой.
  • Знакопеременная группа разрешима тогда и только тогда, когда её порядок не больше 4. Точнее, - четверной группе Клейна, а при .
  • Группа имеет представление
здесь .

Примечания

  1. Н. Н. Вильямс. Знакопеременная группа // Математическая энциклопедия : [в 5 т.] / Гл. ред. И. М. Виноградов. — М.: Советская энциклопедия, 1979. — Т. 2: Д — Коо. — 1104 стб. : ил. — 150 000 экз.


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