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In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is
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, specifically group theory, a subgroup series of a group is a chain of subgroups:
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.
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 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, specifically group theory, an abnormal subgroup is a subgroup H of a group G such that for all x in G, x lies in the subgroup generated by H and H x, where H x denotes the conjugate subgroup xHx−1.
In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.
In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups.
In mathematics, in the field of group theory, a subgroup of a group is termed seminormal if there is a subgroup such that , and for any proper subgroup of , is a proper subgroup of .
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 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 operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator.
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, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G.
Tuval Shmuel Foguel is Professor of Mathematics at Adelphi University in Garden City, New York. Tuval Foguel was born in 1959 in Berkeley, California to Hava and Shaul Foguel and he is a descendant of Saul Wahl. Through his mother Hava, Professor Foguel is related to Nahum Sokolow. Professor Foguel received his B.S. in mathematics from York College, City University of New York in 1988 and his PhD in Mathematics under Michio Suzuki from the University of Illinois at Urbana–Champaign in 1992 with a focus on finite groups. He has introduced the term conjugate-permutable subgroup. In the past, Professor Foguel has also taught at the University of the West Indies, North Dakota State University, Auburn University Montgomery, and Western Carolina University.
References
- ↑ Foguel, Tuval (1997), "Conjugate-permutable subgroups", Journal of Algebra, 191 (1): 235–239, doi:10.1006/jabr.1996.6924, MR 1444498 .