Minimal normal subgroup

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Symbol-free definition

A nontrivial subgroup of a group is termed a minimal normal subgroup if it is normal and the only normal subgroup properly contained inside it is the trivial subgroup.

Definition with symbols

A nontrivial subgroup $H$ of a group $G$ is termed a minimal normal subgroup if it is normal and for any normal subgroup $K$ of $G$ such that $K\leq H$ , either $K=H$ or $K$ is trivial.

Formalisms

In terms of the minimal operator

This property is obtained by applying the minimal operator to the property: nontrivial normal subgroup
View other properties obtained by applying the minimal operator

Examples

In the group $\mathbb {Z}$ of integers, there are no minimal normal subgroups. That's because every number is a factor of some bigger number.
In the group of integers mod $n$ , i.e. the cyclic group of order $n$ , there is a unique minimal normal subgroup of order $p$ for each prime divisor $p$ of $n$ .
In the symmetric group on three letters, the cyclic subgroup of order three (generated by a 3-cycle) is a minimal normal subgroup. The cyclic subgroup of order two is not a normal subgroup.
In a nilpotent group, any minimal normal subgroup must actually be a minimal subgroup (i.e. it must be cyclic of prime order), that's because in a nilpotent group, any nontrivial normal subgroup intersects the center nontrivially.
The subgroup comprising double transpositions and the identity, inside the symmetric group on four letters, is an example of a minimal normal subgroup that is not minimal as a subgroup (i.e., it contains proper nontrivial subgroups). This subgroup is abstractly isomorphic to the Klein four-group.

Relation with other properties

Stronger properties

Metaproperties

Left realization

A group can be realized as a minimal normal subgroup of another group if and only if it is characteristically simple, i.e. it has no proper nontrivial characteristic subgroups. Thus, for instance, a cyclic group of order four can never be realized as a minimal normal subgroup in any group, whereas the Klein four-group can.

When the whole group is a solvable, this further forces every minimal normal subgroup to be elementary Abelian.

For full proof, refer: Minimal normal implies characteristically simple, minimal normal implies elementary Abelian in finite solvable

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
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A minimal normal subgroup of a minimal normal subgroup cannot be normal unless the minimal normal subgroup is a simple normal subgroup.

Intersection-triviality

This subgroup property is intersection-trivial, viz the intersection of two distinct subgroups satisfying the property must be trivial

The intersection of a minimal normal subgroup with any normal subgroup is either itself or trivial. In particular, the intersection of two distinct minimal normal subgroups is trivial. Thus, the property of being minimal normal is intersection-trivial.

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:MinimalNormalSubgroups
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
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References

Textbook references

Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info}, Page 25 (definition in paragraph, immediately suceeding definition of characteristically simple group)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info}, Page 93 (definition in paragraph)
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 106 (no definition given; term implicitly introduced in exercises)