Minimal normal subgroup

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 $$K \le H$$, either $$K=H$$ or $$K$$ is trivial.

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.

Stronger properties

 * Weaker than::Simple normal subgroup
 * Weaker than::Normal subgroup of prime order

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 $$\{ e \} = H_0 \le H_1 \le \dots H_n = G$$ such that each $$H_i$$ is normal in $$G$$ and $$H_{i+1}/H_i$$ is a minimal normal subgroup of $$G/H_i$$.

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.

Other facts

 * Minimal normal subgroup and core-free maximal subgroup need not be permutable complements
 * Plinth theorem

Textbook references

 * , Page 25 (definition in paragraph, immediately suceeding definition of characteristically simple group)
 * , Page 93 (definition in paragraph)
 * , Page 106 (no definition given; term implicitly introduced in exercises)