next up previous
�U�@��: p-Group �W�@��: Cauchy's Theorem �e�@��: �ҩ� Cauchy's Theorem �ҥΪ�

Cauchy �w�z

�ڭ̲{�b�Ϋe�����Ъ� group action �ҩ� Cauchy's Theorem. �A���j�իe���� G �èS���n�D abelian, �ҥH�ڭ̪������A�Ω�@�몺 group.

Theorem 4.2.1 (Cauchy's Theorem)   �Y G �O�@�� group �B p �㰣 G ���Ӽ�, �䤤 p �O�@�ӽ��, �h�s�b a $ \in$ G ���� ord(a) = p.

�� ��. �ڭ̧Q�Ϋe�����Ъ� group action, �o�̧ڭ̥O m = p, H �O Sp ���� (1  2   ...   p) �o�@�� p-cycle �Ҳ��ͪ� cyclic subgroup. ��

S = {(a1,..., ap) $\displaystyle \in$ Gp | a1 . a2 ... ap = e}.

�Y | G| = n �Q�Ϋe�����l (4.2) �� | S| = np - 1, �G�Ѱ��] p | n �o p �㰣 | S|. �]�N�O��

| S| $\displaystyle \equiv$ 0(mod p) (4.5)

�� lemma 3.4.7 �� (1  2  ...  p) �o�@�� p-cycle �� order �� p, �G�� | H| = p. �]�N�O�� H �O�@�� p-group. �]���Q�� Proposition 4.1.4 �M���l (4.5) ��

| S0| $\displaystyle \equiv$ | S| $\displaystyle \equiv$ 0(mod p).

�]�N�O�� p �㰣 | S0|. ���L�Ѧ��l (4.4) �� | S0|$ \ge$1, �A�[�W p �㰣 | S0|, �]�N�O | S0| �O p �����ƥB���O 0. �]���ڭ̪�

| S| > 1.

���y�ܻ� S0 �����F�w���� (e, e,..., e) �o�Ӥ����~�٦���L������. �Ѧ��l (4.3), �ڭ̪��D�b�o�Ǥ������O (a, a,..., a) �o�اΦ�, �B ap = e. �]���o a$ \ne$e �B ap = e, �]�N�O�� ord(a) = p. $ \qedsymbol$

�^�U�@�U�q�e�ڭ̥��ҩ��F�b abelian group ���ΤU�� Cauchy �w�z, �A�Q�Υ��ұo abelian group �� Sylow �w�z. �N�ӧڭ̤]�|�γo�@�� group �� Cauchy �w�z�ҩ��@�� group �� Sylow �w�z.

next up previous
�U�@��: p-Group �W�@��: Cauchy's Theorem �e�@��: �ҩ� Cauchy's Theorem �ҥΪ�
Administrator 2005-06-18