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# 抽象代數-conjugate&normal subgroup

1.

Suppose that a is conjugate to b in a group G.

Show that C(a) is also conjugate to C(b) in G

其中C(a)={ g屬於G | ga=ag }

2.

Show that if H and K are normal subgroups of a group G such that

H聯集K={e} , then hk=kh for all h屬於H and k屬於K

Update:

To 淡淡:

這樣不是只有證明b和C(a) conjugate嗎？

還是其實已經證C(a)和C(b) conjugate了...？

可以說明一下嗎?謝謝~

Update 2:

!!!

我看懂了

### 3 Answers

Rating

- no nicknameLv 51 decade agoFavorite Answer
1. a conjugate to b in G <=> there exists g in G such that g a g^-1 = b

for all A in c(a) , i.e. aA=Aa, try to show gAg^-1 commute with b.

gAg^-1 b = g A (g^-1 b g) g^-1 = g A (a ) g^-1 = g a A g^-1

= g a g^-1 g A g^-1 =b g A g^-1. so we done.

- CopestoneLv 41 decade ago
第二題是交集，不是聯集，即 H ∩ K = {e}。

你的問題都很基本，不知道你遇到什麼困難？

2009-12-21 02:06:43 補充：

我沒什麼時間，如果幾天後還沒有人給你回答，我就來處理一下喔。

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