Seminormal subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
This is a variation of permutability|Find other variations of permutability |

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