Very generic question about Commutator and Center Let $G$ be a finite group and $G'$ the commutator group of $G$.


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*What can I say about $G' \cap Z(G)$? 


Could you be as specific as possible about p-Groups?
 A: Well, this is sort of a broad question, but here a couple random facts.


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*$G'\cap Z(G)\leqslant \Phi(G)$ for all groups $G$.

*Special $p$-groups are an interesting case of $p$-groups in which $G'\cap Z(G)= Z(G)=G'$.
There is also some kind of theorem about $[G:G'\cap Z(G)]^2$ dividing the degrees of a group's irreducible characters, which I believe I saw in Huppert, but I cannot remember the exact statement at the moment.
A: There is a well-known theorem (I think due to Grün) which asserts that if $P$ is a Sylow $p$-subgroup of a finite group $G,$ then $P \cap G^{\prime} \cap Z(G) \leq P^{\prime}.$ If $G$ itself is a $p$-group, then I am not sure how much you can expect to say. In that case, if $G$ is non-Abelian, then $G^{\prime} \cap Z(G)$ is always non-trivial, but it may well have order $p.$ On the other hand, when $p$ is odd, there is always an $n$-generator finite $p$-group $G$ of nilpotent class $2$ and exponent $p$ such that $G^{\prime} = Z(G)$ is elementary Abelian of order $p^{\frac{n(n-1)}{2}}$ and $[G:G^{\prime}] = p^{n}.$
