# Isologic groups

From Groupprops

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

## Definition

Let be a subvariety of the variety of groups. Two groups and (*not necessarily in *) are termed **isologic** with respect to if there exists an isologism between them with respect to .

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

## Facts

- Any group in 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 | fixed-class isologic groups |

variety of all groups | any two groups are isologic with this relation (this is the coarsest possible notion of isologism) |