Isologic groups

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

• Any group in ${\mathcal {V}}$ is isologic to the trivial group.
• Isologism with respect to variety is isologism with respect to any bigger variety
• Isologic groups with respect to fixed nilpotency class lower than theirs have equal nilpotency class

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)