Homoclinism of groups

Definition

For any group $G$, let $\operatorname {Inn} (G)$ denote the inner automorphism group of $G$, $G'$ denote the derived subgroup of $G$, and $Z(G)$ denote the center of $G$.

Let $\omega _{G}$ denote the map from $\operatorname {Inn} (G)\times \operatorname {Inn} (G)$ to $G'$ defined by first taking the map $G\times G\to G'$ 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 $(\zeta ,\varphi )$ where $\zeta$ is a homomorphism from $\operatorname {Inn} (G_{1})$ to $\operatorname {Inn} (G_{2})$ and $\varphi$ is a homomorphism from $G_{1}'$ to $G_{2}'$ such that $\varphi \circ \omega _{G_{1}}=\omega _{G_{2}}\circ (\zeta \times \zeta )$. In symbols, this means that for any $x,y\in \operatorname {Inn} (G_{1})$ (possibly equal, possibly distinct), we have:

$\varphi (\omega _{G_{1}}(x,y))=\omega _{G_{2}}(\zeta (x),\zeta (y))$

Pictorially, the following diagram must commute:

${\begin{array}{ccc}\operatorname {Inn} (G_{1})\times \operatorname {Inn} (G_{1})&{\stackrel {\zeta \times \zeta }{\to }}&\operatorname {Inn} (G_{2})\times \operatorname {Inn} (G_{2})\\\downarrow ^{\omega _{G_{1}}}&&\downarrow ^{\omega _{G_{2}}}\\G_{1}'&{\stackrel {\varphi }{\to }}&G_{2}'\\\end{array}}$

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.