Subgroup properties

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 subnormalsubgroup 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. (Wikipedia).

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Related pages

Conjugate-permutable subgroup | Transitive relation | Finitely generated group | Subgroup | Transitive closure | Ascendant subgroup | Normal subgroup | Mathematics | Characteristic subgroup | Pronormal subgroup | Quasinormal subgroup | Sylow subgroup | Nilpotent group | T-group (mathematics) | Group theory | Descendant subgroup | Group (mathematics) | Serial subgroup