Maximal normal subgroup

Symbol-free definition
A defining ingredient::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 defining ingredient::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
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 $$H$$.

Stronger properties

 * Normal subgroup whose quotient is an absolutely simple group
 * Weaker than::Subgroup of index two
 * Weaker than::Normal subgroup of prime index

Weaker properties

 * Normal subgroup

Other related properties

 * Minimal 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.

Textbook references

 * , Page 90 (definition introduced in paragraph)
 * , Page 24, Exercises 1.4, Problem 6 (no definition introduced, but problem implicitly asks to show the equivalence of the two definitions given)