Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.

In notation, $H$ is $k$ -subnormal in $G$ if there are subgroups

H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G

of $G$ such that $H_{i}$ is normal in $H_{{i+1}}$ for each $i$ .

A subnormal subgroup is a subgroup that is $k$ -subnormal for some positive integer $k$ . Some facts about subnormal subgroups:

A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.
Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal.
Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.
Every 2-subnormal subgroup is a conjugate-permutable subgroup.

The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.

If every subnormal subgroup of G is normal in G, then G is called a T-group.

References

Robinson, Derek J.S. (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
Ballester-Bolinches, Adolfo; Esteban-Romero, Ramon; Asaad, Mohamed (2010), Products of Finite Groups, Walter de Gruyter, ISBN 978-3-11-022061-2

Subnormal subgroup

See also

References