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, 
is 
-subnormal in 
if there are subgroups
of such that 
is normal in 
for each 
.
A subnormal subgroup is a subgroup that is -subnormal for some positive integer .
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.
See also
- Characteristic subgroup
- Normal core
- Normal closure
- Ascendant subgroup
- Descendant subgroup
- Serial subgroup