Isoclinic groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Two groups are said to be isoclinic if there is an isoclinism between them, i.e., there are isomorphisms between their inner automorphism groups as well as isomorphisms between their derived subgroups such that the isomorphisms are compatible with the commutator map Failed to parse (syntax error): {\displaystyle \operatorname{Inn}(G) \times \operatorname{Inn(G) \to G'} .

Facts

Groups isoclinic to the trivial group

A group is isoclinic to the trivial group if and ony if it is abelian. In that case, the inner automorphism group and commutator subgroup are both trivial, and thus the isomorphisms are just the trivial maps.

Subgroups isoclinic to each other

Any subgroup of a group is isoclinic to its product with the center of the group. In particular, this means that any two subgroups having nonempty intersection with the same cosets of the center of the whole group are isoclinic.

In particular, any cocentral subgroup of a group is isoclinic to the whole group.