# Maximal subgroup of group of prime power order

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: maximal subgroup with a group property imposed on theambient group: group of prime power order

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

Suppose is is a group of prime power order (with underlying prime ) and is a subgroup of . We say that is a **maximal subgroup of group of prime power order** if is a group of prime power order and satisfies any of the following equivalent conditions:

- is a maximal subgroup of .
- is a maximal normal subgroup of .
- is a normal maximal subgroup of .
- is a subgroup of prime index in .
- is a normal subgroup of and the quotient group is a group of prime power order.
- contains the Frattini subgroup of and the quotient is a codimension one subspace in the Frattini quotient , viewed as a vector space over the field of elements.

### Equivalence of definitions

See equivalence of definitions of maximal subgroup of group of prime power order. The key ingredients to the proof are prime power order implies nilpotent, nilpotent implies every maximal subgroup is normal, and equivalence of definitions of group of prime order (which shows that any simple abelian group must be cyclic of prime order).