Minimal normal subgroup

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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

Related notions

Socle is the join of all minimal normal subgroups in a group. For a nilpotent p-group, the socle is Omega-1 of the center, see socle equals Omega-1 of center in nilpotent p-group.
A monolithic group is a group that contains a unique minimal normal subgroup that is contained in every nontrivial normal subgroup. The minimal normal subgroup in this case is termed a monolith and it also coincides with the socle.
Chief series is a normal series where each successive quotient is a minimal normal subgroup in the quotient of the whole group by the lower end. In other words, it is a series $\{e\}=H_{0}\leq H_{1}\leq \dots H_{n}=G$ such that each $H_{i}$ is normal in $G$ and $H_{i+1}/H_{i}$ is a minimal normal subgroup of $G/H_{i}$ .

Facts

General facts

Minimal normal implies characteristically simple: Any minimal normal subgroup must be a characteristically simple group, i.e., it must have no proper nontrivial characteristic subgroups. This follows from the fact that characteristic of normal implies normal.
Characteristically simple and normal fully normalized implies minimal normal: In particular, any characteristically simple group is minimal normal in its holomorph. This is related to the fact that left transiter of normal is characteristic.

Facts in certain kinds of ambient groups

Statement	Condition on ambient group	Conclusion about minimal normal subgroup
Minimal normal implies elementary abelian in finite solvable group	finite solvable group	elementary abelian group, i.e., a direct product of one or more copies of a group of prime order.
Minimal normal implies additive group of a field in solvable group	solvable group	additive group of a field (either an elementary abelian group or a vector space over the rationals)
Minimal normal implies central in nilpotent group	nilpotent group	central subgroup. Further, we can conclude that it is a group of prime order, because central implies normal, so any minimal normal subgroup in the center must be a minimal subgroup, and hence cyclic of prime order.
Minimal normal implies contained in Omega-1 of center for nilpotent p-group, or equivalently, socle equals Omega-1 of center in nilpotent p-group	nilpotent p-group; finite p-groups are always nilpotent, so any group of prime power order	contained in the first omega subgroup of the center.
Minimal normal implies pi-group or pi'-group in pi-separable group	pi-separable group	either a $\pi$ -group (all prime factors from the prime set $\pi$ ) or a $\pi '$ -group (all prime factors from outside the prime set $\pi$ ).

Additional conditions and conclusions

Statement	Additional condition	Conclusion
minimal normal subgroup with order not dividing index is characteristic, minimal normal subgroup with order greater than index is characteristic	the order of the subgroup does not divide its index	it is a characteristic subgroup
normality-large and minimal normal implies monolith	normality-large subgroup: intersection with every nontrivial normal subgroup is normal	monolith, i.e., it is contained in every nontrivial normal subgroup, so the group is a monolithic group
self-centralizing and minimal normal implies monolith (this follows from the previous fact and the fact that self-centralizing and normal implies normality-large)	self-centralizing subgroup: contains its own centralizer in the whole group	monolith, i.e., it is contained in every nontrivial normal subgroup, so the group is a monolithic group

Other facts

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
Learn more about using GAP

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)