# Finite-pi-potentially verbal subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Suppose is a finite group and is a subgroup of . We say that is a **finite-pi-potentially verbal subgroup** of if the following holds: There exists a group containing such that all prime factors of the order of also divide the order of and is a verbal subgroup of .

## Relation with other properties

### Stronger properties

- Central subgroup of finite group
- Cyclic normal subgroup of finite group
- Homocyclic normal subgroup of finite group