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

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:

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.

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