Subnormal subgroup

In group theory, a subgroup of a group is called a Subnormalteiler (or subnormal subgroup ) if a Subnormalreihe from to exist, that is, if there is a finite chain ≤ ... ≤ of subsets of such that each normal subgroup of is.

Subnormalteiler were - still under the name nachinvariante subgroup - for the first time by Helmut Wielandt published in his 1939 habilitation dissertation considered a generalization of the invariant subgroups. Wielandt showed among other things that in finite groups, the product of two Subnormalteiler is always another subnormal, ie the Subnormalteiler form a federation.

The concept of Subnormalteilers is so far a generalization of the notion of normal subgroup, does not have to be normal in the whole group as a Subnormalteiler necessarily. Every normal subgroup but always a Subnormalteiler.

Example

The subgroup generated by a reflection of the symmetric group is a normal subgroup of the Klein four-group, which in turn is normal in. So Subnormalteiler of, but not a normal subgroup, there is not in.