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.