Seminormal subgroup

History
The term seminormal subgroup has many equivalent definitions. The one given here is due to Xiang Ying Su.

Definition with symbols
A subgroup $$A$$ of a finite group $$G$$ is termed seminormal if there exists a subgroup $$B$$ such that $$AB = G$$ and for any proper subgroup $$C$$ of $$B$$, $$AC$$ is a proper subgroup of $$G$$.

Such a $$B$$ is termed a S-supplement to $$A$$. The set of all S-supplements of a group $$A$$ is denoted as $$S(A)$$.

The set of S-supplements is closed under conjugacy
It turns out that if $$B \in S(A)$$ then so does $$B^x$$ for any $$x \in G$$. This result was proved by Su in 1988.

Stronger properties
These properties are stronger in the case of finite groups:


 * Normal subgroup
 * Permutable subgroup
 * Subgroup of prime index

Relation with simplicity
In a simple group, any proper nontrivial seminormal subgroup must be a subgroup of prime index.