Maximal subgroup of group of prime power order

This article describes a property that arises as the conjunction of a subgroup property: maximal subgroup with a group property imposed on the ambient 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

Definition

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

1. $H$ is a maximal subgroup of $G$.
2. $H$ is a maximal normal subgroup of $G$.
3. $H$ is a normal maximal subgroup of $G$.
4. $H$ is a subgroup of prime index in $G$.
5. $H$ is a normal subgroup of $G$ and the quotient group $G/H$ is a group of prime power order.
6. $H$ contains the Frattini subgroup $\Phi (G)$ of $G$ and the quotient $H/\Phi (G)$ is a codimension one subspace in the Frattini quotient $G/\Phi (G)$, viewed as a vector space over the field of $p$ 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).

Relation with other properties

Weaker properties

• Maximal subgroup of finite nilpotent group
• Subgroup of prime index
• Normal subgroup of prime index
• Maximal normal subgroup
• Isomorph-normal subgroup
• Order-normal subgroup