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 .
This definition of seminormal subgroups is due to Xiang Ying Su. [1] [2]
Every normal subgroup is seminormal. For finite groups, every quasinormal subgroup is seminormal.
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.
In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element while being associative as well as having an identity element and inverse elements. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The formulation of the axioms is, however, detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
In mathematics, specifically group theory, a nilpotent groupG is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}.
In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
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 group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.
In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997 and arose in the context of the proof that for finite groups, every quasinormal subgroup is a subnormal subgroup.
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.
The Schur–Zassenhaus theorem is a theorem in group theory which states that if is a finite group, and is a normal subgroup whose order is coprime to the order of the quotient group , then is a semidirect product of and . An alternative statement of the theorem is that any normal Hall subgroup of a finite group has a complement in . Moreover if either or is solvable then the Schur–Zassenhaus theorem also states that all complements of in are conjugate. The assumption that either or is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.
Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group with those of its p-local subgroups, that is to say, the normalizers of its non-trivial p-subgroups.
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.
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.
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups.
In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space. This notion generalizes the notions of a hyperbolic group and of a relatively hyperbolic group and includes a significantly wider class of examples, such as mapping class groups and Out(Fn).
