Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $H,K$ be subgroups of a group $G$. If $[H,K]$ is finite and $H$ is finitely generated, then the centralizer $C_K(H)$ has finite index in $K$. More precisely, if $[H,K]$ has order $n$ and $H$ is $m$-generated, then $|K : C_K(H)| \leq n^m$.

share|cite|improve this question

1 Answer 1

up vote 3 down vote accepted

Hint 1: If $H = \langle x_1, \ldots, x_m \rangle$, then $C_K(H) = \cap_{i=1}^m C_K(x_i)$.

Hint 2: $[K : C_K(x)]$ is the number of conjugates $k^{-1}xk$, where $k \in K$.

Hint 3: Notice that $k^{-1}xk = x[x,k]$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.