Cocentral subgroup

Symbol-free definition
A subgroup of a group is termed a cocentral subgroup if it satisfies the following equivalent conditions:


 * 1) Its product with the center is the whole group.
 * 2) The inclusion map from the subgroup to the group induces maps between the corresponding inner automorphism groups and derived subgroup that give rise to an isoclinism.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed cocentral if $$HZ(G) = G$$ where $$Z(G)$$ denotes the center of $$G$$.

Dichotomy
For a group with nontrivial center, any maximal subgroup of a group is either cocentral or contains the center of the group. This idea is of use in proving that every maximal subgroup of a nilpotent group is normal. It is also related to the fact that upper join-closure of cocentrality is central factor.

Trimness
The property of being cocentral is an, that is, every group is cocentral as a subgroup of itself. However, it is not in general trivially true. In fact, the trivial subgroup is cocentral if and only if the group is abelian.

If $$H$$ is cocentral in $$G$$, $$H$$ is also cocentral in any intermediate subgroup $$K$$. This follows from the fact that $$Z(K)$$ contains $$K \cap Z(G)$$.

If $$H \le K \le G$$, with $$H$$ a cocentral subgroup of $$K$$ and $$K$$ a cocentral subgroup of $$G$$, then $$H$$ is cocentral in $$G$$.

If $$H \le K \le G$$ with $$H$$ a cocentral subgroup of $$G$$, $$K$$ is also a cocentral subgroup of $$G$$.

It is possible to have a group $$G$$ and two cocentral subgroups $$H_1$$ and $$H_2$$ of $$G$$ such that $$H_1 \cap H_2$$ is not a cocentral subgroup of $$G$$.

It is possible to have a subgroup $$H$$ of a group $$G$$ that is a concentral subgroup in each of two intermediate subgroups $$K_1$$ and $$K_2$$ but not in the join $$\langle K_1, K_2 \rangle$$.