�ڭ̴��I�L���ǯS�� order �� group, �ڭ̥i�H�ȥѨ� order �N��P�_�X�o�� group ���ƻ�ˤl (�Ҧp order p �� group �O cyclic, order p2 �� group �O abelian). �{�b�ڭ̭n�ͧ�h���������G.
���] G �� r �� Sylow p-subgroup. �ѲĤT Sylow �w�z (Theorem
4.6.1) �� r | q �B r = pk + 1. ���L�Y r1, ����
r
p + 1 > q, �o�M r | q �ۥ٬�. �]���o r = 1, �G�� G �� Sylow
p-group �O G �� normal subgroup.
�ڭ̱��U�Ӭ� n = 1 �����p.
P �M Q �J�M���O normal subgroup, �p�G���ҩ�
P Q = {e}
���ܥ� Theorem 3.2.4 �i�o
G
P×Q. �M�� P
Q
�P�ɬO P �M Q �� subgroup (Lemma 1.5.1), �G�� Lagrange �w�z
(Theorem 2.2.2) �� | P
Q| �P�ɾ㰣 | P| = p �M | Q| = q.
�]���o
| P
Q| = 1, �]�N�O��
P
Q = {e}.
�n�F�ڭ̪�
G P×Q, �M�� | P| = p, | Q| = q ���O���, �G��
Corollary 2.2.3 �M Theorem 3.1.1 ��
P
/p
�B
Q
/q
. �]���Q�� Lemma 3.2.5 �M Corollary
3.2.3 �o
�p�G
q | p - 1 ����? �ڭ̨Ӭݳ�²�檺 q = 2 �����p. �]�N�O��
| G| = 2p, �䤤 p �O�@�ө_���. ���ɥ� Proposition 4.7.1 ��,
G ���ߤ@�� Sylow p-subgroup P �O G �� normal subgroup.
�S�ѩ� | P| = p, �G�� Corollary 2.2.3 ���s�b a G �B
ord(a) = p �ϱo
P =
a
. �] 2 �㰣 | G|, �Q��
Cauchy �w�z (Theorem 4.2.1) ���s�b b
G �B
ord(b) = 2.
(�`�N: ���ɦ] Lagrange's Theorem ��
b
P �B
P
b
= {e}.) �� P �O G �� normal subgroup ���s�b
i
, �ϱo
b . a . b-1 = ai. �ڭ̭n���D i �O�h��. �ѩ�
�Y i = pk + 1, ����
b . a = a . b. �M��
a
b
= {e}. �� Lemma 3.4.8 ��
ord(a . b) = 2p = | G|. ���y�ܻ� G �O�@�� cyclic group.
�Y i = pk - 1, ����
b . a = a-1 . b, ��
a-1a (�]
ord(a) = p
2), �G�� G ���O abelian. �Y�O
B =
b
,
�] P �O G �� normal subgroup, �ѲĤG Isomorphism �w�z (Theorem
2.6.4) �� P . B �O G ���@�� subgroup, �B
Proposition 4.7.3 �i�D�ڭ� Dp �O�ߤ@�� order �� 2p ��
nonabelian group. �ƹ�W��Ҧ��� n3, ���s�b�@�� nonabelian
group
�̫�ڭ̷Q�H���Q�Ҧ� order �p�� 10 �� group �����ǧ@�� group �o�ӳ���������.
���M order �� 1 ���N�u�� identity. order �� 2, 3, 5, 7 ��
group ���O cyclic ���O isomorphic to,
/2
,
/3
,
/5
,
/7
.
order �� 4 �� group �� Proposition 4.3.3 �������, ���O
isomorphic to
/4
�M
/2
×
/2
. �P�z order ��
9 ���]�u�����, ���O isomorphic to
/9
�M
/3
×
/3
.
order �� 6 �� group �� Proposition 4.7.3 ���]�����, �@�ӬO
abelian �t�@�ӬO nonabelian, ���̤��O isomorphic to
/6
�M
D3. �����P�ǩγ\�|�ð�: �ڭ̾ǹL S3 ���]�� 3! = 6 �Ӥ���,
����S���C�X�O? �O��i! �ƹ�W S3 �O nonabelian, �ڭ̥i�H�ұo
S3
D3. �䤤 S3 ��
(1 2 3) �N������ D3 ����
order �� 3 ������ a, �� (1 2) �N������ D3 ���� order ��
2 ������ b, �B
�̫ᦳ�I�Ƥ⪺�O order �� 8 �� group. Abelian �������٦n�B�z,
�ڭ̪��D��
/8
,
/4
×
/2
�M
/2
×
/2
×
/2
. �ܩ� nonabelian
�����ڭ̤w������
�{�� a G �䤤
ord(a) = 4. �]
a
�O�@�� order
�� 4 = 22 �� subgroup �� | G| = 8 = 23, �G�ѲĤ@ Sylow �w�z (Theorem
4.4.2) ��, G �s�b�@�� subgroup K �䤤
| K| = 22 + 1 �B
a
�O K �� normal subgroup. ���ѩ� | K| = | G|, �G��
K = G. �]��
a
�O K �� normal subgroup.
�J�M
a
�O K �� normal subgroup, ���� b
G ��
b
a
, �Ҧs�b
i
�ϱo
b . a . b-1 = ai. �ڭ̷Q�n���D i ���h��. �����[��Y
ord(b . a . b-1) = r, �Y
�̫�ѩ�
b a
, �u���i��
ord(b) = 2 ��
ord(b) = 4. �Y
ord(b) = 2 �h��
G
D4; �Y
ord(b) = 4, �h��
G
Q8. �G�o�� order 8 �� nonabelian group �u�����.
�p�G�P�Ǧ�������M�i�H�@����U�h: order 11 �� group ���h�� (�o��²��)? order 12 �����h��? �o�ˤ@���U�h���D�V�ӶV�x��. �j�a�������F�Ѱ��D���x���שM order �j�p�L��, �ӬO�M���]�ƪ����Ѧ���. �j�a������|����V�j�N�V����, �Ҧp order 16 �� group �N�� 14 ��, �� order 32 �� group �N�����F 51 �Ӥ��h.