When going about my mathematical business, I often feel my group theoretic shortcomings very sorely. Here is an example:
Let $G$ be a group, $Z(G)$ the centre of $G$, and $\pi: G \to G/Z(G)$ the canonical projection.
Let $H < G$ be a subgroup.
Is it true that $\pi(H) \cong H/Z(H)$?
This is of course a special case of the more general situation in which we want to understand the images of subgroups $H < G$ under the quotient map $\pi: G \to G/N$, where $N \vartriangleleft G$ is a normal subgroup.
I have looked through several textbooks on elementary group theory but have found no allusion to the above.
A reference or a proof/counterexample would be very helpful. Thank you for your attention.
No, this is not the case. $Z(H)$ can contain elements that $Z(G)$ doesn't, because commuting with all of $H$ is less restrictive than commuting with all of $G$. So $Z(H)$ being larger than $Z(G)\cap H$ is not impossible at all.
Take $G=S_5$, for instance. Then $Z(G)=\{e\}$. However, if $H$ is any of several possible non-trivial abelian subgroups, then $Z(H)=H$.