Subnormal subgroup
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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
of G such that Hi is normal in Hi + 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 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, 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. In fact, the relation of subnormality can be defined as the transitive closure of the relation of normality.
[edit] See also
- Normal subgroup
- Characteristic subgroup
- Normal core
- Normal closure
- Ascendant subgroup
- Descendant subgroup
- Serial subgroup
[edit] References
- Robinson, Derek J.S. (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
