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 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

:$H=H\_0,H\_1,H\_2ldots\; H\_k=G$

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 normal subgroup (and vice versa).
* A finite group is a nilpotent group if and only if every subgroup of it 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, in fact, normal. In particular, every 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 subnormalsubgroup is subnormal. In fact, the relation of subnormality can be defined as the transitive closure of the relation of normality.

=See also=

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

References