# Isoclinism of groups

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## History

The notion of isoclinism seems to have been first introduced by Philip Hall mainly for the purpose of classifying finite p-groups, in his 1937 paper.

This page is mostly about the mappings that are used to define isoclinism. For more on the equivalence relation of being isoclinic, see isoclinic groups.

## Definition

### Short definition

An isoclinism is an isologism of groups with respect to the subvariety of abelian groups.

### Full 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 $Z(G) \times Z(G)$.

An isoclinism of groups $G_1$ and $G_2$ is a pair $(\zeta,\varphi)$ where $\zeta$ is an isomorphism of $\operatorname{Inn}(G_1)$ with $\operatorname{Inn}(G_2)$ and $\varphi$ is an isomorphism of $G_1'$ with $G_2'$ such that $\varphi \circ \omega_{G_1} = \omega_{G_2} \circ (\zeta \times \zeta)$. Explicitly, this means that for any $x,y \in \operatorname{Inn}(G_1)$, we have the following: $\varphi(\omega_{G_1}(x,y)) = \omega_{G_2}(\zeta(x),\zeta(y))$

Pictorially, the following diagram must commute: $\begin{array}{ccc} \operatorname{Inn}(G_1) \times \operatorname{Inn}(G_1) & \stackrel{\zeta \times \zeta}{\to} & \operatorname{Inn}(G_2) \times \operatorname{Inn}(G_2) \\ \downarrow^{\omega_{G_1}} & & \downarrow^{\omega_{G_2}}\\ G_1' & \stackrel{\varphi}{\to} & G_2'\\ \end{array}$

Two groups are said to be isoclinic groups if there is an isoclinism between them.

### Definition in terms of homoclinism

An isoclinism is an invertible homoclinism of groups, i.e., a homoclinism for which both the component homomorphisms are isomorphisms. Equivalently, it is an isomorphism in the category of groups with homoclinisms.