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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Does $[ G : C_G(x) ] = [ G/K : C_{G/K}(x) ] [ K : C_K(x) ]$ hold for all finite groups $G$ and $p$-elements $x$, where $K = O_{p'}(G)$ is the largest normal subgroup of $G$ with order coprime to $p$?

A related statement is true: Let $G = H \ltimes K$ be a semidirect product of $H$ with the (arbitrary) normal subgroup $K$, and let $x$ in $H$. Suppose $hk$ centralizes $x$. Then considering the group mod $K$, one gets that $h$ centralizes $x$. Hence $k$ centralizes $x$, and $C_G(x) = C_H(x) \cdot C_K(x)$. Importantly, we have the equality:

$$[ G : C_G(x) ] = [ G/K : C_{G/K}(x) ] [ K : C_K(x) ] .$$

If $G$ splits over $K = O_{p'}(G)$ so that $G = H \ltimes K$, then every $p$-element is conjugate to some element of $H$, and the previous implies the conjugacy class sizes behave as I expect. However, $G$ often does not split over $K$ and I am not sure what happens then.

So suppose $G$ is a finite group with normal subgroup $K$. One still has that every $h$ centralizing $x$ gives rise to an $hK$ in $C_{G/K}(x)$ and of course $C_G(x)$ is a union of cosets of $C_K(x)$. However, it is not clear that all elements $hK$ of $C_{G/K}(x)$ give rise to elements $h$ in $C_G(x)$. In other words, I can only prove that:

$$[ G : C_G(x) ] \geq [ G/K : C_{G/K}(x) ] [ K : C_K(x) ] .$$

I checked for counterexamples (small groups up to order $1000$ except $768$, primitive groups up to degree $500$ except $343$, and perfect groups up to order $10^6$ available in GAP), but found no counterexamples to equality. I am not sure if the coprime hypothesis is truly relevant, but it is the case of interest and I could not think of any other way to say "$x$ in $H$" when $G$ did not split. With absolutely no hypotheses on $x$ or $K$, then $G$ dihedral of order 8 with $K = Z(G)$ and $x$ a non-central involution gives a counterexample, but this is not at all similar to the case I am interested in.

share|cite|improve this question
up vote 3 down vote accepted

If $H$ is a p-subgroup of $G$, then it is true that: $$[G:C_G(H)]=[G/K:C_{G/K}(H)]\ [K:C_K(H)].$$

This follows from two facts:

$C_K(H) = C_G(H)\cap K$; this is clear.

$ C_{G/K}(H)=C_G(H)K/K$. To prove this, note first that $N_{G/K}(H)=N_G(H)K/K$. This is equivalent to $N_G(HK)=N_G(H)K$, and this follows from the Frattini argument applied to $HK$ in $N_G(HK)$. Now the centralizer case follows because $N_G(H)$ is mapped to $N_{G/K}(H)$, $H$ is mapped to an isomorphic copy of itself, and the quotient map respects the action of $N_G(H)$ on $H$ (so, in particular, respects the kernel of this action, $C_G(H)$).

share|cite|improve this answer
Thanks, looks good. I think this shows the coprime hypothesis is a good one, though Sylow and centralizer are stronger than needed. I think this is more or less the proof that the fusion system of G and G/K are isomorphic, but kept nicely inside G. – Jack Schmidt May 15 '11 at 7:49

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.