When is the center a direct summand?

Question

Suppose $G$ is a group. When do we have a decomposition $$G\cong Z(G)\oplus H$$ for some group $H$?

Answer 1

If $G\cong Z(G)\oplus H$, then we have a projection map $\pi:G\to Z(G)$ which is the identity on $Z(G)$. Conversely, given a projection map $\pi:G\to Z(G)$ which is the identity on $Z(G)$, then let $H=\ker \pi$. Then we have $G\cong Z(G) \times H$, since $Z(G)H=G$, $Z(G)\cap H = 1$, and $Z(G)$ commutes with everything in $H$.

So $G\cong Z(G)\oplus H$ for some $H$ if and only if there is a map $\pi:G\to Z(G)$ such that $\pi|_{Z(G)}=1_{Z(G)}$.

An alternative equivalent condition is that if $H$ is a subgroup of $G$ such that $Z(G)H=G$, and $Z(G)\cap H =1$, then $H$ is automatically normal, and $G\cong Z(G)\oplus H$. The proof here is that $Z(G)$ provides a system of representatives for $G/H$, and if $z\in Z(G)$, then $zH=Hz$, so all left cosets of $H$ are right cosets of $H$ as well. Thus $H$ is normal.

As Arturo Magidin says in the comments, subgroups $H$ such that $HZ(G)=G$ are called cocentral.