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�U�@��: RING �W�@��: �i�� Group ���ʽ� �e�@��: Sylow p-subgroups ���Ӽ�


Sylow �w�z������

�ڭ̤w�j�P���Ч��F group ���@�ǰ򥻩ʽ�. �b�o���� group ���̫�@�`���ڭ̤��Ф@�ǥi�H�Q�� Sylow �w�z�o�쪺�ʽ�. ���o�ǩʽ褣�u�n�Ψ� Sylow �w�z, �ٻݭn�@�ǫe���ǹL���w�z���U, �ҥH�⥦��b group ���̫�@�`���j�a�Ʋߤ@�U�e���ҾǪ�, �]�⵹ group �@�ӧ���������.

�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.

Proposition 4.7.1   �Y G �O�@�� group �B | G| = pnq, �䤤 n$ \ge$1, p �M q �O�۲���ƥB p > q. �h G �� Sylow p-group �O G �� normal subgroup.

�� ��. �ڭ̥u���D group �� order, �S����L���T��, �ҥH���D���i��� normal ���w�q�Ӫ����ұo���w�z. �۫H�j�a�|�Q�� Second Sylow's Theorem �a. �p�G�ڭ̯��ұo G ���� Sylow p-subgroup �u���@��, ����Q�βĤG Sylow �w�z (Corollary 4.5.2) �N�i�����O G �� normal subgroup �F.

���] G �� r �� Sylow p-subgroup. �ѲĤT Sylow �w�z (Theorem 4.6.1) �� r | q �B r = pk + 1. ���L�Y r$ \ne$1, ���� r$ \ge$p + 1 > q, �o�M r | q �ۥ٬�. �]���o r = 1, �G�� G �� Sylow p-group �O G �� normal subgroup. $ \qedsymbol$

�ڭ̱��U�Ӭ� n = 1 �����p.

Proposition 4.7.2   �Y G �O�@�� group �B | G| = pq, �䤤 p �M q �O�۲���ƥB p > q. �Y�S�� q$ \nmid$p - 1, �h G �O�@�� cyclic group.

�� ��. �o�N���������ҩ��F. �����ѩ� p, q �ҬO���, Cauchy �w�z (Theorem 4.2.1) �i�D�ڭ� G ������� subgroups P �M Q �� order ���O�� p �M q. ��� P �O G �� Sylow p-subgroup, Q �O Sylow q-subgroup. �� Proposition 4.7.1 �� P �O G �� normal subgroup, �� Q �O? ���] G ���� r �� Sylow q-subgroup. �ѲĤT Sylow �w�z (Theorem 4.6.1) �� r | p �B r = qk + 1. �p�G r$ \ne$1, �� r | p �� r = p, �]���o p = qk + 1. �]�N�O qk = p - 1. ���M q$ \nmid$p - 1 �ۥ٬�. �G�� r = 1, �]�]���ѲĤG Sylow �w�z (Corollary 4.5.2) �� Q �]�O G �� normal subgroup.

P �M Q �J�M���O normal subgroup, �p�G���ҩ� P $ \cap$ Q = {e} ���ܥ� Theorem 3.2.4 �i�o G $ \simeq$ P×Q. �M�� P $ \cap$ Q �P�ɬO P �M Q �� subgroup (Lemma 1.5.1), �G�� Lagrange �w�z (Theorem 2.2.2) �� | P $ \cap$ Q| �P�ɾ㰣 | P| = p �M | Q| = q. �]���o | P $ \cap$ Q| = 1, �]�N�O�� P $ \cap$ Q = {e}.

�n�F�ڭ̪� G $ \simeq$ P×Q, �M�� | P| = p, | Q| = q ���O���, �G�� Corollary 2.2.3 �M Theorem 3.1.1 �� P $ \simeq$ $ \mathbb {Z}$/p$ \mathbb {Z}$ �B Q $ \simeq$ $ \mathbb {Z}$/q$ \mathbb {Z}$. �]���Q�� Lemma 3.2.5 �M Corollary 3.2.3 �o

G $\displaystyle \simeq$ P×Q $\displaystyle \simeq$ $\displaystyle \mathbb {Z}$/p$\displaystyle \mathbb {Z}$×$\displaystyle \mathbb {Z}$/q$\displaystyle \mathbb {Z}$ $\displaystyle \simeq$ $\displaystyle \mathbb {Z}$/pq$\displaystyle \mathbb {Z}$.

�G�o G �O�@�� cyclic group. $ \qedsymbol$

�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 $ \in$ G �B ord(a) = p �ϱo P = $ \langle$a$ \rangle$. �] 2 �㰣 | G|, �Q�� Cauchy �w�z (Theorem 4.2.1) ���s�b b $ \in$ G �B ord(b) = 2. (�`�N: ���ɦ] Lagrange's Theorem �� b $ \not\in$P �B P $ \cap$ $ \langle$b$ \rangle$ = {e}.) �� P �O G �� normal subgroup ���s�b i $ \in$ $ \mathbb {N}$, �ϱo b . a . b-1 = ai. �ڭ̭n���D i �O�h��. �ѩ�

b . (b . a . b-1) . b-1 = b . ai . b-1 = (b . a . b-1)i = ai2.

�� b2 = (b-1)2 = e, �G a = ai2. �]�N�O�� ai2 - 1 = e. �Q�� Lemma 2.3.2 �o ord(a) = p | i2 - 1. �ѩ� p �O���, �ڭ̱o p | i - 1 �� p | i + 1. �]�N�O�� i = pk + 1 �� i = pk - 1.

�Y i = pk + 1, ���� b . a = a . b. �M�� $ \langle$a$ \rangle$ $ \cap$ $ \langle$b$ \rangle$ = {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-1$ \ne$a (�] ord(a) = p$ \ne$2), �G�� G ���O abelian. �Y�O B = $ \langle$b$ \rangle$, �] P �O G �� normal subgroup, �ѲĤG Isomorphism �w�z (Theorem 2.6.4) �� P . B �O G ���@�� subgroup, �B

P . B/P $\displaystyle \simeq$ B/P $\displaystyle \cap$ B.

�ѩ� P $ \cap$ B = {e}, �� | P . B| = | P| . | B| = 2p. �]�N�O�� P . B = G. ���y�ܻ�

G = {ai . bj | 0$\displaystyle \le$i$\displaystyle \le$p - 1, 0$\displaystyle \le$j$\displaystyle \le$1}.

�ƹ�W�ڭ̥i�ҩ��s�b�o�˪��@�� group. �ڭ̺٤��� dihedral group of degree p, �O�@ Dp. ��X�H�W�ڭ��ұo�F�H�U�����G.

Proposition 4.7.3   �Y G �O�@�� group �B | G| = 2p, �䤤 p �O�@�ө_���, �h

G $\displaystyle \simeq$ $\displaystyle \mathbb {Z}$/2p$\displaystyle \mathbb {Z}$    or    G $\displaystyle \simeq$ Dp.

Proposition 4.7.3 �i�D�ڭ� Dp �O�ߤ@�� order �� 2p �� nonabelian group. �ƹ�W��Ҧ��� n$ \ge$3, ���s�b�@�� nonabelian group

Dn = {ai . bj | 0$\displaystyle \le$i$\displaystyle \le$n - 1, 0$\displaystyle \le$j$\displaystyle \le$1},

�O�Ѩ�Ӥ��� a, b �Ҳ���, �䤤 ord(a) = n, ord(b) = 2 �B b . a = a-1 . b. �o�˪� nonabelian group, �ڭ̺٤��� dihedral group of degree n. ���� order �� 2n. ���L�� n ���O��Ʈ�, Dn �N���@�w�O�ߤ@�� order �� 2n �� nonabelian group �F.

�̫�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, $ \mathbb {Z}$/2$ \mathbb {Z}$, $ \mathbb {Z}$/3$ \mathbb {Z}$, $ \mathbb {Z}$/5$ \mathbb {Z}$, $ \mathbb {Z}$/7$ \mathbb {Z}$.

order �� 4 �� group �� Proposition 4.3.3 �������, ���O isomorphic to $ \mathbb {Z}$/4$ \mathbb {Z}$ �M $ \mathbb {Z}$/2$ \mathbb {Z}$×$ \mathbb {Z}$/2$ \mathbb {Z}$. �P�z order �� 9 ���]�u�����, ���O isomorphic to $ \mathbb {Z}$/9$ \mathbb {Z}$ �M $ \mathbb {Z}$/3$ \mathbb {Z}$×$ \mathbb {Z}$/3$ \mathbb {Z}$.

order �� 6 �� group �� Proposition 4.7.3 ���]�����, �@�ӬO abelian �t�@�ӬO nonabelian, ���̤��O isomorphic to $ \mathbb {Z}$/6$ \mathbb {Z}$ �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 $ \simeq$ D3. �䤤 S3 �� (1  2  3) �N������ D3 ���� order �� 3 ������ a, �� (1  2) �N������ D3 ���� order �� 2 ������ b, �B

(1  2)(1  2  3) = (2  3) = (3  2  1)(1  2).

�P�z order �� 10 �� group �]�����, ���̤��O isomorphic to $ \mathbb {Z}$/10$ \mathbb {Z}$ �M D5.

�̫ᦳ�I�Ƥ⪺�O order �� 8 �� group. Abelian �������٦n�B�z, �ڭ̪��D�� $ \mathbb {Z}$/8$ \mathbb {Z}$, $ \mathbb {Z}$/4$ \mathbb {Z}$×$ \mathbb {Z}$/2$ \mathbb {Z}$ �M $ \mathbb {Z}$/2$ \mathbb {Z}$×$ \mathbb {Z}$/2$ \mathbb {Z}$×$ \mathbb {Z}$/2$ \mathbb {Z}$. �ܩ� nonabelian �����ڭ̤w������

D4 = {ai . bj | 0$\displaystyle \le$i$\displaystyle \le$3, 0$\displaystyle \le$j$\displaystyle \le$1},

�䤤 ord(a) = 4, ord(b) = 2 �B b . a = a-1 . b. �ƹ�W�٦��t�@�ӫܱ`���� order �� 8 �� nonabelian group Q8, �٤��� quaternion group. �̱`���� Q8 ���ܪk�p�U:

Q8 = {±1, ±i, ±j ±k},

�䤤 i2 = j2 = k2 = - 1 �B i . j = - j . i = k. �]�� Q8 �� order �� 4 �������� 6 �� (�Y ±i, ±j �M ±k), �� D4 ���u����� (�Y a �M a3) �G�� Q8 �M D4 �ä� isomorphic. �ڭ̭n�ҩ� order 8 �� nonabelian group �u���o���.

Proposition 4.7.4   �Y G �O�@�� order �� 8 �� nonabelian group, �h

G $\displaystyle \simeq$ D4    or    G $\displaystyle \simeq$ Q8.

�� ��. �] | G| = 8, �� Lagrange �w�z (Corollary 2.3.4) �� G �������� order �u��O 1, 2, 4 �� 8. �ڭ̭n�ҩ� G �������@������ order �� 4. �Y G ���������� order �� 8, �� G �� cyclic �M G �O nonabelian �ۥ٬�. �]���Y�S������ order �� 4 ���ܥ��N G ���D identity �������� order �Ҭ� 2, �]�N�O���Ҧ��� g $ \in$ G ������ g2 = e. �Y�u�p��, ���� a, b $ \in$ G, �ڭ̪�

e = (a . b)2 = (a . b) . (a . b),

�G�o

a . b = a . (a . b) . (a . b) . b = b . a

�o�S�M G �O nonabelian �ۥ٬�. �G�� G �����s�b order �� 4 ������.

�{�� a $ \in$ G �䤤 ord(a) = 4. �] $ \langle$a$ \rangle$ �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 $ \langle$a$ \rangle$ �O K �� normal subgroup. ���ѩ� | K| = | G|, �G�� K = G. �]�� $ \langle$a$ \rangle$ �O K �� normal subgroup.

�J�M $ \langle$a$ \rangle$ �O K �� normal subgroup, ���� b $ \in$ G �� b $ \not\in$$ \langle$a$ \rangle$, �Ҧs�b i $ \in$ $ \mathbb {N}$ �ϱo b . a . b-1 = ai. �ڭ̷Q�n���D i ���h��. �����[��Y ord(b . a . b-1) = r, �Y

(b . a . b-1)r = b . ar . b-1 = e,

�h�o ar = e. �G�� Lemma 2.3.2 �� 4 | r. �M�� (b . a . b-1)4 = b . a4 . b-1 = e �G�� r | 4. �]�N�O�� ord(b . a . b-1) = 4. �� Lemma 2.3.3 ���u���� i = 1, 3 �� ord(ai) = 4, �G���Y b . a . b-1 = ai, �h i = 1 �� i = 3. ���L�p�G i = 1 ���� b . a = a . b �G�� G �O abelian, ���S�M G �O nonabelian �����]�ۥ٬�. �]����

b . a = a3 . b = a-1 . b.

�̫�ѩ� b $ \not\in$$ \langle$a$ \rangle$, �u���i�� ord(b) = 2 �� ord(b) = 4. �Y ord(b) = 2 �h�� G $ \simeq$ D4; �Y ord(b) = 4, �h�� G $ \simeq$ Q8. �G�o�� order 8 �� nonabelian group �u�����. $ \qedsymbol$

�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.


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�U�@��: RING �W�@��: �i�� Group ���ʽ� �e�@��: Sylow p-subgroups ���Ӽ�
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