Seminormal subgroup

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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

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History

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

Definition

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 $S(A)$ .

Facts

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. Template:Fill-proof

Relation with other properties

Stronger properties

These properties are stronger in the case of finite groups:

Relation with simplicity

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

References

On Seminormal subgroups by Tuval Foguel, Journal of Algebra, 165 (1994), Page 633-635
Normality conditions on subgroups of finite simple groups by Tuval Foguel,

External links

Tuval Foguel's publications which include copies of his articles on seminormal subgroups