# Math Help - Centralizer of a Group

1. ## Centralizer of a Group

Let G be a group. Let the centralixer C(X), X a subset of G, be defined as follows: C(X)={g in G | gx=xg for all x in X}.

Prove the following:

If X is a subset of Y, then C(Y) is a subset of C(X)

X is a subset of C(C(X))

C(X) = C(C(C(X)))

2. a) Suppose $g\in C(Y)$. We want to show $g\in C(X)$ i.e. $gx=xg,\forall x\in X$.
Take $x\in X\stackrel{X\subset Y}{\Longrightarrow}x\in Y\stackrel{g\in C(Y)}{\Longrightarrow}gx=xg$.
b) Suppose $x\in X$. We need $x\in C(C(X))=\{g\in G | gy=yg,\forall y\in C(X)\}$, so we want to prove that $\forall y\in C(X)$ we have $xy=yx$.
Take $y\in C(X)\stackrel{x\in X}{\Longrightarrow}yx=xy$.
c)" $\subset$" Suppose $g\in C(X)$. We want $g\in C(C(C(X)))$ i.e. $gy=yg,\forall y\in C(C(X))$. Take $y\in C(C(X))\stackrel{g\in C(X)}{\Longrightarrow}yg=gy$.
" $\supset$": From b) we know $X\subset C(C(X))\stackrel{\text{a)}}{\Longrightarrow}C(C(C( X)))\subset C(X)$.