Maximal normal subgroup

This is a subgroup property obtained by applying the maximal operator to the trim subgroup property: normal subgroup

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

Symbol-free definition

A proper subgroup of a group is termed a maximal normal subgroup if it is normal and the only normal subgroup properly containing it is the whole group.

Definition with symbols

A proper subgroup $H$ of a group $G$ is termed a maximal normal subgroup if it is normal and for any normal subgroup $K$ of $G$ such that $K$ ≥ $H$, either $K=H$ or $K=G$.

In terms of the maximal operator

The property of being a maximal normal subgroup is an element of the subgroup property space, and is obtained by applying the maximal operator to the property of normality.

Stronger properties

• Normal subgroup whose quotient is an absolutely simple group

Weaker properties

• Normal subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

A maximal normal subgroup of a maximal normal subgroup need not be normal. Even if it is, it is clearly not a maximal normal subgroup (because there's a bigger normal subgroup containing it).

Quotient-realization

The quotient of any group by a maximal normal subgroup is simple. In fact, the quotient of a group by a subgroup is simple if and only if that subgroup is a maximal normal subgroup.