# Finite group having a subgroup series with prime indexes

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **finite group having a subgroup series with prime indices** is a finite group with a subgroup series starting at the trivial subgroup and terminating at the whole group, where each subgroup has prime index in its successor.

In other words, it is a finite group whose max-length equals the sum of exponents of its prime divisors in the prime factorization of its order.

The smallest finite group that does *not* have this property is the alternating group of degree six.

## Relation with other properties

### Stronger properties

- Finite cyclic group
- Finite abelian group
- Finite nilpotent group
- Finite supersolvable group
- Finite solvable group

## Metaproperties

### Normal subgroups

This group property is normal subgroup-closed: any normal subgroup (and hence, any subnormal subgroup) of a group with the property, also has the property

View other normal subgroup-closed group properties

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property

View other finite direct product-closed group properties