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.