# Prove C(a) is a subgroup of G

• Mar 24th 2010, 09:40 AM
nikie1o2
Prove C(a) is a subgroup of G
Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.
• Mar 24th 2010, 10:18 AM
tonio
Quote:

Originally Posted by nikie1o2
Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.

What've you tried? Prove that the group's unit is contained in $C(a)$ and also that $a,b\in C(a)\Longrightarrow ab^{-1}\in C(a)$

Tonio
• Mar 24th 2010, 10:20 AM
hatsoff
Quote:

Originally Posted by nikie1o2
Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.

Let $x,y\in C(a)$. Then $x=axa^{-1}$ and $y=aya^{-1}$. So $xy=axa^{-1}aya^{-1}=axya^{-1}$, and $xy\in C(a)$. Now observe that $(ax^{-1}a^{-1})x=(ax^{-1}a^{-1})(axa^{-1})=e$. So $x^{-1}=ax^{-1}a^{-1}$, that is, $x^{-1}\in C(a)$. Since clearly $e\in C(a)$, then it follows that $C(a)$ is a subgroup of $G$