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 $H$.

Examples

In a nilpotent group, any maximal subgroup is normal, hence it is a maximal normal subgroup and the quotient is a cyclic group of prime order. For full proof, refer: Nilpotent implies every maximal subgroup is normal

Thus, any maximal subgroup in a nilpotent group gives an example of a maximal normal subgroup. For instance, in the cyclic group of four elements, any subgroup of order two, is a maximal normal subgroup. Similarly, in a dihedral group of order $2n$, the cyclic subgroup of order $n$ is a maximal normal subgroup.

Any subgroup of index two, or any normal subgroup of prime index is a maximal normal subgroup. For instance, in the symmetric group on three letters, while the subgroup of order three is a maximal normal subgroup. In solvable groups, the converse is also true: any maximal normal subgroup is of prime index (though, not every maximal subgroup is normal).

In a non-solvable group, there could be maximal normal subgroups whose index is not prime, and whose quotient is 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.

Infinite groups need not always 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

Stronger properties

• Normal subgroup whose quotient is an absolutely simple group
• Subgroup of index two
• Normal subgroup of prime index

Weaker properties

• Normal subgroup

Related group properties

• Group in which every maximal subgroup is normal: All nilpotent groups satisfy this property
• Group in which every maximal normal subgroup has prime index: All solvable groups satisfy this property

Related notions

• Composition series is a subnormal series where each member is a maximal normal subgroup of the adjacent member bigger than it. The quotients for a composition series, called the composition factors, are simple groups. The Jordan-Holder theorem guarantees uniqueness of the composition factors up to permutation and isomorphism classes.
• A one-headed group is a group with a unique maximal normal subgroup. Such a maximal normal subgroup is termed a head.
• The Jacobson radical is defined as the intersection of all maximal normal subgroups. This is related to, but different from, the Frattini subgroup, which is the intersection of all maximal subgroups. In nilpotent groups, and in particular in groups of prime power order, the two notions coincide.

Facts

• Nilpotent implies every maximal subgroup is normal
• Maximal normal implies prime index in solvable

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
Learn more about using GAP

References

Textbook references

• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 90 (definition introduced in paragraph)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info}, Page 24, Exercises 1.4, Problem 6 (no definition introduced, but problem implicitly asks to show the equivalence of the two definitions given)