What does $Z_2(S)$ mean where $S$ is a group.

Question

In an article by Bob Oliver "Simple fusion systems over p-groups with abelian subgroup of index p" in Notation 2.2., he defines a set $Z_2 = Z_2(S)$ where $S$ is a nonabelian $p$-group. The problem is that I don't know what $Z_2(S)$ means. I know $Z(S)$ is the center, but I have never seen $Z_2(S)$. Can somebody help me?

Answer 1

As i.m. soloveichik points out in the comments, $Z_2(G)$ presumably refers to the second center of the group $G$, i.e. the second term in the upper central series.

This is indeed a subgroup of $G$, and it has several equivalent definitions:

• It is the inverse image of $Z\bigl(G/Z(G)\bigr)$ under the quotient homomorphism $G \to G/Z(G)$.

• It is the set $\{g\in G \mid [g,h]\in Z(G)\text{ for all }h\in G\}$.

• It is the set $\{g\in G \mid [[g,h],k]=1\text{ for all }h,k\in G\}$.

Note that $Z_2(G)$ is a normal (and indeed characteristic) subgroup of $G$, and that $Z_2(G)$ always contains $Z(G)$. Moreover, $Z_2(G)$ is equal to $G$ if and only if $G$ is nilpotent of class 2, i.e. if and only if $G$ is a central extension of an abelian group.