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, ${\displaystyle H}$ is ${\displaystyle k}$-subnormal in ${\displaystyle G}$ if there are subgroups

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

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

A subnormal subgroup is a subgroup that is ${\displaystyle k}$-subnormal for some positive integer ${\displaystyle k}$. 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.
• A subgroup ${\displaystyle H}$ is subnormal in ${\displaystyle G}$ if and only if ${\displaystyle |HK|}$ divides ${\displaystyle |G|}$ for every subgroup ${\displaystyle K}$ of ${\displaystyle G}$.

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.

See also

• Characteristic subgroup
• Normal core
• Normal closure
• Ascendant subgroup
• Descendant subgroup
• Serial subgroup

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