Let $G$ be a group, and $H \le G$. For what pairs $(G,H)$ is $Z(H)=Z(G)\cap H$ true?

Question

Let $G$ be a group and $H$ be a subgroup of $G$. Classify all pairs $(G,H)$ such that $Z(H)=Z(G)\cap H$ is true.

My attempt:

If $G$ is abelian then $H \le G$ will also be abelian. Hence $Z(H)=H$ and $Z(G)=G$. So $Z(H)=H=G\cap H=Z(G)\cap H$. Also for any group $G$, $H=\{1\}$will also hold trivially. If $H=G$, then $Z(G)=Z(G)\cap G$ is also always true for any $G$, since $Z(G) \le G$

But this is not true for all pairs, consider the quaternion group $G = Q_8$, then $Z(Q_8) = \{1,-1\} $ , and $H = <i> = \{1,-1,-i,i\}$, since $H$ is cyclic, this implies $Z(H) = H $. But $H \ne \{ 1,-1\} \cap H = \{1,-1\}$. A non-trivial example for which this is true is $G = GL_n(\mathbb{F})$ and $H = SL_n(\mathbb{F})$. $\, Z(SL_n(\mathbb{F})) = Z(GL_n(\mathbb{F}))\cap SL_n(\mathbb{F})$ = All scalar matrices with determinant $1$.

Is there a way to classify all these pair of groups ? Are there finitely examples which doesn't satisfy this ?

Answer 1

This is true for example in the following cases. If (a) $G=HZ(G)$, or (b) $H \unlhd G$ and $H \cap G'=1$ (in (b) note that $H \subseteq Z(G)$). This can be generalized by using the concept of isoclinism between groups (write $\sim$), an equivalence relation on the class of groups coarser than isomorphism (see for instance my paper here). In case $G$ is finite, then your property is true whenever (a) $G \sim H$ or (b) $H \unlhd G$ and $G \sim G/H$.