Conjugation as a group action

�U�@��: p-group ��ʽ� �W�@��: p-Group �e�@��: p-Group

Conjugation as a group action

�^�U�@�U�ڭ̴��L�Y�T�w x $\in$ G, ��N�� g $\in$ G, g^.x^.g^-1 �٬� x ��@�� conjugation. �ƹ�W�o�O G �� S = G ��@�� group action.

�Y G �O�@�� group. �O S = G, �Ӷȧ� S �ݦ��O�@�Ӷ��X. �Ҽ{ G �� S ��@�Φp�U: ��N�� a $\in$ G, x $\in$ S, �ڭ̩w�q a*x = a^.x^.a^-1.

�ڭ̭n�ҩ��o�� (G, S,*) �O�@�� group action. ��ˬd (Act1). �Y a $\in$ G, x $\in$ S, �h a*x = a^.x^.a^-1. �] a, x, a^-1 �Ҧb G �� G �O�@�� group, �G a^.x^.a^-1 $\in$ G = S. �o�� a*x $\in$ S. �A�Ӧ] e*x = e^.x^.e^-1 = x, �G�� (Act2) �]�ŦX. �̫�Y a, b $\in$ G, x $\in$ S, �h

a*(b*x) = a*(b^.x^.b^-1) = a^.(b^.x^.b^-1)^.a^-1,

�M��

(a^.b)*x = (a^.b)^.x^.(a^.b)^-1 = (a^.b)^.x^.(b^{-1 .}a^-1).

�G�ѵ��X�v�� a*(b*x) = (a^.b)*x, �o�� (Act3).

�b�o�� action ��] S = G, �G�۵M�� | S| = | G|. �{�b�Ӭ� S₀ �O��? �өw�q�Y x $\in$ S₀ ��ܹ�Ҧ�� g $\in$ G �Ҧ� g*x = x. �]�N�O�� x, ��N�� g $\in$ G, �Ҷ��ŦX g^.x^.g^-1 = x. �Ѧ��o g^.x = x^.g, $\forall$ g $\in$ G. ��y�ܻ� S₀ ��һݩM�Ҧ� G ��i�洫. �Ϥ��Y x $\in$ S �i�H�M G ��Ҧ��洫��, �h

g*x = g^.x^.g^-1 = x^.g^.g^-1 = x,

�]�N�O�� x $\in$ S₀.

�p�G�j�a��Ѫ��, �ڭ̴��b 1.4 �`��гo�˪��Ҧ��X Z(G) �٬� G �� center, �B�Q�� Lemma 1.5.1 ��L Z(G) �O�@�� G �� subgroup. �`��, �ڭ��ұo�F

S₀ = Z(G) = {x $\displaystyle \in$ G | g^.x = x^.g, $\displaystyle \forall$ g $\displaystyle \in$ G}.

(4.6)

�̫�ڭ��٬O�n�j�զ]�w�� e $\in$ Z(G), �G��

| S₀| $\displaystyle \ge$ 1.

(4.7)

�U�@��: p-group ��ʽ� �W�@��: p-Group �e�@��: p-Group

Administrator 2005-06-18