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

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 properties
VIEW 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 $p$, the prime-cube order group:U(3,p), which is the unique non-abelian group of order $p^3$ and exponent $p$. All its maximal subgroups are elementary abelian of order $p^2$.
• SmallGroup(16,4) is a group of order $16$, where all the maximal subgroups are isomorphic to direct product of Z4 and Z2.