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 $\gamma _{G}$ 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 $G$ and $H$ is a pair $(\zeta ,\phi )$ where $\zeta$ is a homomorphism from $\operatorname {Inn} (G)$ to $\operatorname {Inn} (H)$ and $\varphi$ is a homomorphism from $G'$ to $H'$ such that $\varphi \circ \gamma _{G}=\gamma _{H}\circ (\zeta \times \zeta )$.

Facts

• Homomorphism induces homoclinism

Related notions

• Isoclinism is an invertible homoclinism.