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Encyclopedia > 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 chain of subgroups of the group, each one normal in the next, beginning at G and ending at H. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Group theory is that branch of mathematics concerned with the study of groups. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

In notation, H is subnormal in G if there are

such that Hi is normal in Hi + 1 for each i.

A finite group is a nilpotent group if and only if every subgroup of it is subnormal. In mathematics, a finite group is a group which has finitely many elements. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...

Every quasinormal subgroup, and, more generally, every conjugate permutable subgroup, of a finite group is subnormal. In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup. ...

Every pronormal subgroup that is also subnormal, is, in fact, normal. In particular, every Sylow subgroup is normal if and only if it is normal. The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the...

The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. In fact, subnormality can be defined as the transitive closure of normality. In grammar, a verb is transitive if it takes an object. ... Results from FactBites:

 PlanetMath: subnormal subgroup (108 words) Subnormality is a strictly weaker condition than normality, as normality of subgroups is not transitive. In a nilpotent group, all subgroups are subnormal. This is version 18 of subnormal subgroup, born on 2002-12-14, modified 2006-11-29.
 Reference.com/Encyclopedia/Subnormal subgroup (213 words) 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 chain of subgroups of the group, each one normal in the next, beginning at H and ending at G. A 1-subnormal subgroup is a normal subgroup (and vice versa). The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal.
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