Sylow p-subgroups ��Ӽ�

Theorem 4.6.1 (Third Sylow's Theorem) �Y G �O�@�� group �B | G| = pⁿm, �䤤 n $\ge$ 1, p �O�@�ӽ�ƥB p $\nmid$ m. �O r �� G ��Ҧ� Sylow p-subgroup ��Ӽ�, �h

(1) r | m; (2) r $\displaystyle \equiv$ 1(mod p).

�� . (1) �ڭ̧Q�βĤ@�� group action (G, S,*) ��ҩ� r | m. �Ѧ��l (4.19) ��: �� P' $\in$ S, �ڭ̦� r = | G|/| N(P')|. ��L Lemma 4.4.1 �i�D�ڭ� P' �O N(P') �� subgroup. �ѩ� | P'| = pⁿ, Lagrange �w�z�i�D�ڭ� | N(P')| �O pⁿ ��, �S�ѩ� N(P') �O G �� subgroup, �A�Τ@�� Lagrange �w�z�o | N(P')| = pⁿd �䤤 d | m. �G��

r = $\displaystyle {\frac{\vert G\vert}{\vert N(P')\vert}}$ = $\displaystyle {\frac{p^nm}{p^nd}}$ = $\displaystyle {\frac{m}{d}}$ .

�]�� r | m.

(2) �ڭ̧Q�βĤG�� group action (P, S,*) ��ҩ� r $\equiv$ 1(mod p). �] P �O�@�� p-group, �� Proposition 4.1.4 ��

r = | S| $\displaystyle \equiv$ | S₀|(mod p).

(4.22)

�ڭ̲{�b�ӭp�� | S₀|. �Ѧ��l (4.20) ��Y P' $\in$ S₀ �h P $\subseteq$ N(P'). ��L�e��w�� | N(P')| = pⁿd, �䤤 d | m. �M�ӥ� p $\nmid$ m �� p $\nmid$ d, �]�� | P| = pⁿ �� P �O N(P') ��@�� Sylow p-subgroup. �t�@�譱, �� Lemma 4.4.1 ��, P' �O N(P') �� normal subgroup. �� P' �]�O N(P') �� Sylow p-subgroup. Corollary 4.5.2 �i�D�ڭ� P' �O N(P') �ߤ@�� Sylow p-subgroup. �G�o P = P'. ��y�ܻ� S₀ ��u�i�঳ P �o�Ӥ��. �]��Ѧ��l (4.21) �� S₀ = {P}, �]�N�O�� S₀ �u��@�Ӥ��. �G�Ѧ��l (4.22) �o r $\equiv$ 1(mod p). $\qedsymbol$

Example 4.6.2 (1) �ڭ̪��D�b A₄ �� Sylow 3-subgroup �ä��ߤ@ (Example 4.5.3), �� A₄ �쩳��h�֭� Sylow 3-subgroup �O? ��]�� r ��, �ѩ� | A₄| = 4^.3, �ѲĤT Sylow �w�z (Theorem 4.6.1) �� r | 4 �B r = 3k + 1. �]�N�O r = 1 �� r = 4. �ѩ�w�� r $\ne$ 1, �G�o r = 4. �ƹ�W�b A₄ ��

(1 2 3), (1 2 4), (1 3 4), (2 3 4)

�o�|�� 3-cycles �ӧO��ͪ� cyclic group �Ҭ۲�, �G��o�ǴN�O�Ҧ� A₄ �� Sylow 3-subgroup.

(2) �b A₅ ��h�� Sylow 5-subgroups �O? ��]�� r ��, �ѩ� | A₅| = 5!/2 = 5^.12, �ѲĤT Sylow �w�z (Theorem 4.6.1) �� r | 12 �B r = 5k + 1. �]�N�O r = 1 �� r = 6. �ѩ�w�� A₅ �O simple (Theorem 3.4.26) �ҥH A₅ �� Sylow 5-subgroup ��i��O A₅ �� normal subgroup. �]��ѲĤG Sylow �w�z (Corollary 4.5.2) �� r $\ne$ 1. �G�o r = 6. �ƹ�W�b A₅ ��Ҧ�� 5-cycle �@�� 4! = 24 �� (��ƻ�O? �o�O��ƦC�զX��ӤH��઺��D�a!). ��L��@�� 5-cycle �Ҳ��ͪ� cyclic group �� 4 �� 5-cycle �X�{. �Ҧp:

$\displaystyle \langle$ (1 2 3 4 5) $\displaystyle \rangle$ = {(1 2 3 4 5),(1 3 5 2 4),(1 4 2 5 3),(1 5 4 3 2), I}

�]��o 24 �� 5 cycle �u�� 24/4 = 6 �Ӭ۲�� order 5 �� subgroup. �o�N�O�Ҧ� A₅ �� Sylow 5-subgroup.

�j�a��n�Q Example 4.6.2 �~��. �ĤT Sylow �w�z�ä��O�U�F��, �@��ӻ��ä��Ѥ@�� group �� order �A�Q�βĤT Sylow �w�z�N��X��h�֭� Sylow p-subgroup. ��ɭn�[�J�ҦҼ{�� group ��ʽ�, �Ҧp�b A₅ �� Sylow 2-subgroup �u�� Third Sylow's Theorem �Ӻ�N��i�঳ 3, 5 �� 15 ��. �ҥH�n�i�@�h��Ҷq�~�i��X�u��X��.

Sylow p-subgroups ���Ӽ�

Sylow p-subgroups ��Ӽ�