Seminormal subgroup
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Contents
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 of a finite group is termed seminormal if there exists a subgroup such that and for any proper subgroup of , is a proper subgroup of .
Such a is termed a S-supplement to . The set of all S-supplements of a group is denoted as .
Facts
The set of S-supplements is closed under conjugacy
It turns out that if then so does for any . 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