1=7

����� IA No.7����

����ͤ���Ȥ���Ʊ��뤬�Ĥ��Ƥޤ��ΤǤ����� Ʊ���ϡ�Ʊ�ʹط��ˤ�륯�饹ʬ�����Ѥ��뤳�ȤǸ�뤳�Ȥ��Ǥ���ΤǤ�����

�Է��Ρ���ʬ���ˤ�뺸��;�ླྀ���

$G$ �򷲡�$H$ �򤽤���ʬ���Ȥ��롣���ΤȤ���

$\bullet$ $G$ �ˡ����Τ褦�ˤ���Ʊ�ʹط� $\sim$ ����ޤ롣

$$x\sim y \ {\Leftrightarrow}\$$   ï¿½ï¿½ï¿½ï¿½ $h&isin#in;H$ �����äơ�$$xh=y$$ ������Ω�ġ�

���� 7.1   $G$ �ò·²¡ï¿½$H$ �ò¤½¤ï¿½ï¿½ï¿½Ê¬ï¿½ï¿½ï¿½È¤ï¿½ï¿½ë¡£ï¿½ï¿½ï¿½Î¤È¤ï¿½ï¿½ï¿½ $G$ �ˡ����Τ褦�ˤ���Ʊ�ʹط� $\sim$ ����ޤ롣

$$x\sim y \ {\Leftrightarrow}\$$   ï¿½ï¿½ï¿½ï¿½ $h&isin#in;H$ �����äơ�$$xh=y$$ ������Ω�ġ�

���줫��Ϥ���Ʊ�ʹط��ˤϵ��� $x \sim y$ ������ˡ�

$$x \equiv y \quad ({\operatorname{mod}}\ H)$$

�Ƚ񤤤ơ���$x$ �� $y$ ��$H$ ��ˡ�Ȥ��ƺ���Ʊ�Ǥ���դȸ������ˤ��롣�ͤ��Ƥ��� ��ʬ�� $H$ �����ΤʤȤ��ˤϡ� $({\operatorname{mod}}\ H)$ ��񤯤ΤϾ�ά�����ɤ���$x$ �Υ��饹 $C(x)$ �� $x$ �� $H$ ��ˡ�Ȥ��뺸��;��ȸ�����

$\bullet$ ��Τ褦�˷�᤿Ʊ�ʹط� $\equiv \ ({\operatorname{mod}}\ H)$ �ˤ�� $G$ �ξ����� $G/\equiv$ �� $G/H$ �Ƚ񤭡�$G$ �� $H$ �ˤ�뺸��;�ླྀ��Ȥ�����

�� 7.1   $({\mbox{${\mathbb{Z}}$}},+)$ �ò¤½¤ï¿½ï¿½ï¿½Ê¬ï¿½ï¿½ $n{\mbox{${\mathbb{Z}}$}}$ �dz��ȡ��������������� �ֳ�ĥ�����ֹ�Ť��ò¤µ¤ì¤¿Äºï¿½ï¿½ï¿½Î½ï¿½ï¿½ï¿½ï¿½ ��Ʊ����Τ� �����롣

���� 7.2   $G$ �ò·²¤È¤ï¿½ï¿½ï¿½$H$ �ò¤½¤ï¿½ï¿½ï¿½Ê¬ï¿½ï¿½ï¿½È¤ï¿½ï¿½ë¡£ï¿½ï¿½ï¿½Î¤È¤ï¿½ï¿½ï¿½

1. $G$ ��Ǥ�դθ� $g$ ���Ф��ơ�$g$ �� $H$ ��ˡ�Ȥ��뺸��;�� $C(g)$ ��
   $$gH=\{gh;h\in H\}$$
   �Ȱ��פ��롣
2. $C(g) \ (=gH)$ �� $H$ �Ȥϸ��θĿ�����������
3. $G$ �ΰ̿� $\vert G\vert$ ��ͭ�¤ʤ�С�$G/H$ �θ��θĿ� $\char93 (G/H)$ �� $H$ �ΰ̿� $\vert H\vert$ �ȤϤȤ��ͭ�¤ǡ�
   $$\vert G\vert=\char93 (G/H) \vert H\vert$$
   ������Ω�ġ�

����ݡ�������

�Ĥ��Τ�����������򤷤Ʋ򤭤ʤ����� (���¡����ιֵ��ν�λ���ޤǡ�)

(I).

�� $a$ �����������̿��󽽤�ͭ�½�� $C_{20} (=\langle a; a^{20}=e \rangle)$ ��ͤ��ޤ��� $a^4$ ����������� $C_{20}$ ����ʬ�� $H$ �θ��򤹤٤ƽ񤭽Ф��� $G$ ������$H$ ��ˡ�Ȥ��ƺ���Ʊ�դȸ���Ʊ�ʹط��ǤɤΤ褦�����̤���뤫��$C_{20}$ �θ��Υ��饹ʬ����ɽ���äƼ����ʤ������äˡ�$C_{20}/H$ �θ��θĿ��Ϥ����餫��

(II).

�� $G$ �Ȥ�����ʬ�� $H$ �Ȥ�Ϳ����줿�Ȥ��ޤ���$\vert H\vert$ �� $\char93 (G/H)$ �Ȥ� �Ȥ��ͭ�¤ʤ�С�$\vert G\vert$ ��ͭ�¤Ǥ�����򼨤��ʤ�����