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