# Group of prime power order in which every maximal subgroup is isomorph-free

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 **group of prime power order in which every maximal subgroup is isomorph-free** is a group of prime power order in which every maximal subgroup is an isomorph-free subgroup.

## Examples

- Any cyclic group of prime power order satisfies this property. In fact, a cyclic group of prime power order has a
*unique*maximal subgroup, which has order and equals the first agemo subgroup, i.e., the set of powers. - Among the non-cyclic groups of prime power order, the smallest example of one in which every maximal subgroup is isomorph-free is the semidihedral group of order sixteen (also see subgroup structure of semidihedral group:SD16. This is a group of order having three maximal subgroups of order : a cyclic group, dihedral group and quaternion group.

## Relation with other properties

### Stronger properties

### Weaker properties

- Group of prime power order in which every maximal subgroup is characteristic
- Group in which every maximal subgroup is characteristic