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: * 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. * A subgroup H is subnormal in G if and only if |HK| divides |G| for every subgroup K of G . 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.
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: 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. A subgroup H is subnormal in G if and only if |HK| divides |G| for every subgroup K of G . 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.
See also
Characteristic subgroup Normal core Normal closure Ascendant subgroup Descendant subgroup Serial subgroup
References
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