# Group of prime power order in which all maximal subgroups are isomorphic

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

## Definition

A **group of prime power order in which all maximal subgroups are isomorphic** is a group of prime power order in which all maximal subgroups are isomorphic to each other, i.e., there is (at most) one isomorphism class of maximal subgroups.

## Examples

- Any homocyclic group of prime power order, i.e., a direct product of cyclic groups of the same prime power order, has this property. Examples are elementary abelian groups and cyclic groups.
- The quaternion group is a non-abelian group with this property: all its maximal subgroups are cyclic of order four.
- For odd , the prime-cube order group:U(3,p), which is the unique non-abelian group of order and exponent . All its maximal subgroups are elementary abelian of order .
- SmallGroup(16,4) is a group of order , where all the maximal subgroups are isomorphic to direct product of Z4 and Z2.