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.
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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