Endoclinism-invariant subgroup

As an invariance property
A subgroup $$H$$ of a group $$G$$ is termed an endoclinism-invariant subgroup if it is invariant under any set map $$\sigma:G \to G$$ satisfying all three of these conditions:


 * $$\sigma$$ induces an endomorphism modulo the center, i.e., $$\sigma(xy) = \sigma(x)\sigma(y)$$ mod $$Z(G)$$.
 * $$\sigma$$ restricts to an endomorphism on the derived subgroup, i.e., $$\sigma$$ sends the derived subgroup to itself and the restriction to the derived subgroup is an endomorphism of the derived subgroup.
 * $$\sigma([x,y]) = [\sigma(x),\sigma(y)]$$ for all (possibly equal, possibly distinct) $$x,y \in G$$.

As a two-case property
A subgroup $$H$$ of a group $$G$$ is termed an endoclinism-invariant subgroup if it satisfies either of these conditions:


 * 1) $$H$$ contains the center $$Z(G)$$ and $$H/Z(G)$$ is invariant under any endomorphism of $$G/Z(G)$$ that is the inner automorphism group part of the data specifying an endoclinism.
 * 2) $$H$$ is contained in the derived subgroup $$[G,G]$$ and it is invariant under any endomorphism of $$[G,G]$$ that is the derived subgroup part of the data specifying an endoclinism.

Examples

 * Every group is an endoclinism-invariant subgroup of itself.
 * The trivial subgroup is endoclinism-invariant in every group.
 * All members of the lower central series and derived series are endoclinism-invariant.