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�U�@��: �s�y��h�� subgroups �W�@��: ��� Group ���ʽ� �e�@��: Subgroup


�@�ǯS���� subgroup

�e�����Χڭ̧Ʊ�Q�Τ@�� group �� subgroup �����ڭ̤F�ѳo�@�� group. ���w�@�� group ���F trivial subgroup �~�쩳�n��˧���L�� subgroup �O?���M�F���� group �O�S�� nontrivial proper subgroup ��(�H��ڭ̷|����), �b�o�@�`�ڭ̧Ʊ椶�Ф@�ǥi���� nontrivial proper subgroup ����k.

�� G �O�@�� group ���w a $ \in$ G, �ڭ̧Ʊ�� a �Ӳ��ͤ@�� subgroup. �ܦ۵M���ڭ̪��D a2, a3,..., an,... ���n�b�o�� subgroup ��, �٦� a-1, (a2)-1,...,(an)-1,... �]�n�b�䤤, �̫�O�ѤF e �]�n�b�̭�. �� Corollary 1.2.5, �ڭ̪� (an)-1 = (a-1)n, �ҥH�ڭ̫ܦ۵M���|�w�q�H�U�����X:

$\displaystyle \langle$a$\displaystyle \rangle$ : = {an | n $\displaystyle \in$ $\displaystyle \mathbb {N}$} $\displaystyle \cup$ {(a-1)m | m $\displaystyle \in$ $\displaystyle \mathbb {N}$} $\displaystyle \cup$ {e}.

�ܮe���� Lemma 1.3.3 (�� Lemma 1.3.4) ���D $ \langle$a$ \rangle$ �|�O G ���@�� subgroup. �ڭ̺� $ \langle$a$ \rangle$ �� the cyclic subgroup of G generated by a. ���M�F, �p�G�ڭ̿�� a = e �h $ \langle$a$ \rangle$ = {e} �o�@�� trivial subgroup. �t�@�譱�p�G�ڭ̥i�H���@�� a �ϱo $ \langle$a$ \rangle$ = G, ����ڭ̴N�� G ���@�� cyclic group. �n�`�N���O�ä��O�Ҧ��� group G ���i�H��� a $ \in$ G �ϱo $ \langle$a$ \rangle$ = G.

Example 1.4.1   �ڭ̵��@�Ӧ��ǦP�Ƿ|�d�V���Ҥl. �|�d�V����]�O�e�����L, ���F²�K���]���ڭ̳��� �u . �v �Ӫ��� group ���B��, �ҥH a2 = a . a ���� a �M a �B��⦸, a3 ���ܹB��T��... �̦�����. �{�b�ڭ̦Ҽ{ $ \mathbb {Z}$ �H�[�k�Φ��� group. ���� $ \langle$2$ \rangle$ ���ӬO��˪� subgroup �O? �����ӬO�� 2, 4 = 2 + 2, 6 = 2 + 2 + 2, ...(�d�U�O�d��, ���O�� 2, 4 = 22, 8 = 23,...) �H�� -2, -4, -6, ... ���Ҳզ�. ���y�ܻ��b�� group ���H 2 �ҧΦ��� cyclic subgroup �Y�O�ѩҦ����ƩҲզ���. �t�~�j�a�ܮe���ݥX�� -2 �]�i���ͦP�˪� subgroup. �j�a�]�i�ܮe���ݥX 1 �Ҳ��ͪ� cyclic subgroup �N�O $ \mathbb {Z}$ �����ҥH�ڭ̪��D $ \mathbb {Z}$ �ҧΦ����[�k�s�O�@�� cyclic group.

�j�a���ӳ��ٰO�o, �b�@�몺 group ��, a . b �����o���� b . a. ���L�] a . e = e . a = a �ҥH identity �`�O�M�Ҧ������i�洫��. �ܩ󵹩w�@����, �����Ǥ����i�H�M���洫�O�@�ӫܦ��쪺���D. ���w a $ \in$ G, �ڭ̥i�H�Ҽ{

C(a) = {g $\displaystyle \in$ G | g . a = a . g}.

�o�Ӷ��X�N�O�j�� G ���i�H�M a �洫������. �ڭ̺٤��� the centralizer of a. ���w���N�� a $ \in$ G, �ƹ�W C(a) �|�O G ���@�� subgroup. �Ҧp the centralizer of identity C(e) �N�O G ����.

Proposition 1.4.2   �Y G �O�@�� group �B a $ \in$ G, �h C(a) �O G ���@�� subgroup.

�� ��. �� Lemma 1.3.3, �ڭ̭n�ҩ�: �Y g1, g2 $ \in$ C(a) �h g1 . g2 $ \in$ C(a) �٦� g1-1 $ \in$ C(a). �ƹ�W g1, g2 $ \in$ C(a) �i�D�ڭ� g1 . a = a . g1 �� g2 . a = a . g2. �]��

(g1 . g2) . a = g1 . (g2 . a) = g1 . (a . g2) = (g1 . a) . g2 = (a . g1) . g2 = a . (g1 . g2).

�]�N�O�� g1 . g2 $ \in$ C(a). �t�@�譱, �ѩ� g1 . a = a . g1, �U���W g1-1 �b���䵥�����k��. �ڭ̱o�� (g1 . a) . g1-1 = a. �A���H g1-1 ����䵥��������. �ڭ̱o a . g1-1 = g1-1 . a. �]�N�O�� g1-1 $ \in$ C(a). $ \qedsymbol$

�t�~�@�ر`���� subgroup �O�Ҽ{

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

�ڭ̤@��� Z(G) �� G �� center. �`�N C(a) �O�M G �����S�w���� a �i�洫�������Ҧ������X, �� Z(G) �O�M G ���Ҧ��������i�洫�������Ҧ������X. �ҥH�ڭ̫ܮe���i�ұo

Z(G) = $\displaystyle \bigcap_{a\in G}^{}$C(a).

�������ҩ� C(a) �O G �� subgroup ����k�ڭ̤]�i�H�ҩ� Z(G) �]�O G �� subgroup. �b�o�̧ڭ̤��A���ҩ����L���@�U�ڭ̱N�|�Υt�@�جݪk�ӻ��� Z(G) �O G �� subgroup.


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�U�@��: �s�y��h�� subgroups �W�@��: ��� Group ���ʽ� �e�@��: Subgroup
Administrator 2005-06-18