Isologic groups

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

Definition

Let \mathcal{V} be a subvariety of the variety of groups. Two groups G and H (not necessarily in \mathcal{V}) are termed isologic with respect to \mathcal{V} if there exists an isologism between them with respect to \mathcal{V}.

Being isologic can roughly be thought as being congruent modulo \mathcal{V}, i.e., the difference between the groups lives in \mathcal{V}. In this sense, it behaves like congruence mod n.

Facts

Particular cases

Variety Corresponding notion of isologic groups
variety containing only the trivial subgroup isomorphic groups (this is the finest possible notion of isologism)
variety of abelian groups isoclinic groups
variety of nilpotent groups of class at most c fixed-class isologic groups
variety of all groups any two groups are isologic with this relation (this is the coarsest possible notion of isologism)