�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
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:
a
: = {
an |
n

}

{(
a-1)
m |
m

}

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

�H�[�k���� group. ����

2

���ӬO��˪� subgroup �O? �����ӬO�� 2, 4 = 2 + 2,
6 = 2 + 2 + 2, ...(�d�U�O�d��, ���O�� 2, 4 = 2
2, 8 = 2
3,...)
�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

�����ҥH�ڭ̪��D

�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��. �ܩw�@����,
�����Ǥ����i�H�M���洫�O�@�ӫܦ��쪺���D. ���w a
G,
�ڭ̥i�H�Ҽ{
C(
a) = {
g
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
G, �ƹ�W C(a) �|�O G ���@�� subgroup. �Ҧp
the centralizer of identity C(e) �N�O G ����.
�� ��.
�� Lemma
1.3.3, �ڭ̭n�ҩ�: �Y
g1,
g2
C(
a) �h
g1 . g2
C(
a) �٦�
g1-1
C(
a). �ƹ�W
g1,
g2
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
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
C(
a).
�t�~�@�ر`���� subgroup �O�Ҽ{
Z(
G) = {
g
G |
g . x =
x . g,
x
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) =
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.
�U�@��: �s�y��h�� subgroups
�W�@��: ��� Group ���ʽ�
�e�@��: Subgroup
Administrator
2005-06-18