Subnormal subgroup - Alchetron, The Free Social Encyclopedia

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 , … , 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.

References

Subnormal subgroup Wikipedia

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