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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
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A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroupF of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroupF*, which is generated by the Fitting subgroup and the components of G.
In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups:
In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence
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 with respect to the product of subgroups. The term quasinormal subgroup was introduced by Øystein Ore in 1937.
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In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups,.
In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups. Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.
In mathematics, in the field of group theory, a HN group or hypernormalizing group is a group with the property that the hypernormalizer of any subnormal subgroup is the whole group.
In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of the group.
In mathematics, a group is supersolvable if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.
In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups:
In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.
In the mathematical field of group theory, a subgroup H of a given group G is a serial subgroup of G if there is a chain C of subgroups of G extending from H to G such that for consecutive subgroups X and Y in C, X is a normal subgroup of Y. The relation is written H ser G or H is serial in G.
References
- Dixon, Martyn R. (1994). Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups. World Scientific. p. 6. ISBN 981-02-1795-1.
- Robinson, Derek J.S. (1996). A Course in the Theory of Groups. Springer-Verlag. p. 358. ISBN 0-387-94461-3.
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