Homoclinism of groups

Definition

For any group $G$, let $\mathrm{I}\mathrm{n}\mathrm{n}(G)$ denote the inner automorphism group of $G$, $G^{\prime }$ denote the derived subgroup of $G$, and $Z(G)$ denote the center of $G$.

Let ${\omega }_G$ denote the map from $\mathrm{I}\mathrm{n}\mathrm{n}(G)\times \mathrm{I}\mathrm{n}\mathrm{n}(G)$ to $G^{\prime }$ defined by first taking the map $G\times G\to G^{\prime }$ given as $(x,y)\mapsto x^{-1}{y}^{-1}xy$ and then observing that the map is constant on the cosets of $Z(G)\times Z(G)$.

A homoclinism of groups $G_1$ and $G_2$ is a pair $(\zeta ,\phi )$ where $\zeta$ is a homomorphism from $\mathrm{I}\mathrm{n}\mathrm{n}(G_1)$ to $\mathrm{I}\mathrm{n}\mathrm{n}(G_2)$ and $\phi$ is a homomorphism from ${G_{1^{\prime }}}^{\prime }$ to ${G_{2^{\prime }}}^{\prime }$ such that $\phi \circ {\omega }_{G_1}={\omega }_{G_2}\circ (\zeta \times \zeta )$. In symbols, this means that for any $x,y\in \mathrm{I}\mathrm{n}\mathrm{n}(G_1)$ (possibly equal, possibly distinct), we have:

$\phi ({\omega }_{G_1}(x,y))={\omega }_{G_2}(\zeta (x),\zeta (y))$

Pictorially, the following diagram must commute:

$\begin{array}{ccc}\mathrm{I}\mathrm{n}\mathrm{n}(G_1)\times \mathrm{I}\mathrm{n}\mathrm{n}(G_1)& & \mathrm{I}\mathrm{n}\mathrm{n}(G_2)\times \mathrm{I}\mathrm{n}\mathrm{n}(G_2)\\ {\downarrow }^{{\omega }_{G_1}}& & {\downarrow }^{{\omega }_{G_2}}\\ G_1\text{'}& & G_2\text{'}\\ \end{array}$

Related notions

Term	Meaning
category of groups with homoclinisms	category whose objects are groups and whose morphisms are homoclinisms
isoclinism	an invertible homoclinism, or equivalently, a homoclinism where both the component homomorphisms are isomorphisms
homoclinism of Lie rings	the analogous concept for Lie rings
homologism of groups	a more general concept of which homoclinisms are a special case. Homoclinisms are homologisms with respect to the subvarety of abelian groups in the variety of groups.