# Maximal normal subgroup

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## Definition

### Symbol-free definition

A proper subgroup of a group is termed a maximal normal subgroup if it satisfies the following equivalent conditions:

• It is normal and the only normal subgroup properly containing it is the whole group
• It is normal and the quotient group is a simple group

### Definition with symbols

A proper subgroup $H$ of a group $G$ is termed a maximal normal subgroup if it satisfies the following conditions:

• $H$ is normal in $G$ and for any normal subgroup $K$ of $G$ such that $K \ge H$, either $K=H$ or $K=G$.
• $H$ is normal in $G$ and $G/H$ is a simple group.

### Equivalence of definitions

Further information: Equivalence of definitions of maximal normal subgroup

The equivalence of definitions basically follows from the fourth isomorphism theorem, which establishes a bijection between normal subgroups of $G/H$ and normal subgroups of $G$ containing $H$.

## Facts

Statement Explanation and examples
Nilpotent implies every maximal subgroup is normal In dihedral group:D8, a group of order 8, the cyclic maximal subgroup of order 4 as well as the Klein four-subgroups of dihedral group:D8 are all maximal normal subgroups. Further information: subgroup structure of dihedral group:D8
Index two implies normal In the dihedral group of degree $n$ and order $2n$, the cyclic subgroup of order $n$ is a maximal normal subgroup. In symmetric group:S3, the subgroup A3 in S3 is a maximal normal subgroup. Similarly, in symmetric group:S4, the subgroup A4 in S4 is a maximal normal subgroup.
Subgroup of index equal to least prime divisor of group order is normal For instance, in a group of order 45, any subgroup of index 3 (hence order 15) is maximal normal.
Maximal normal iff normal of prime index in solvable In a solvable group, the maximal normal subgroups are precisely the normal subgroups of prime index. Note that all subgroups of prime index need not be normal, though.
In a non-solvable group, maximal normal subgroups could either be normal of prime index or be normal subgroups with the quotient a simple non-abelian group. For instance, in a simple group such as the alternating group on five letters, the trivial subgroup is a maximal normal subgroup. Similarly, in a special linear group over a finite field such as $SL(n,q)$, the center is a maximal normal subgroup, because the quotient, which is a projective special linear group $PSL(n,q)$, is simple. Further information: special linear group is quasisimple, projective special linear group is simple
Infinite groups need not have maximal normal subgroups For instance, $\mathbb{Q}$, the group of rational numbers, has no maximal normal subgroup.

## Formalisms

### In terms of the maximal operator

This property is obtained by applying the maximal operator to the property: proper normal subgroup
View other properties obtained by applying the maximal operator

## Testing

### GAP command

This subgroup property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for listing all subgroups with this property is:MaximalNormalSubgroups
View subgroup properties testable with built-in GAP command|View subgroup properties for which all subgroups can be listed with built-in GAP commands | View subgroup properties codable in GAP