Real element, the Glossary
In group theory, a discipline within modern algebra, an element x of a group G is called a real element of G if it belongs to the same conjugacy class as its inverse x^, that is, if there is a g in G with x^g.[1]
Table of Contents
14 relations: Brauer–Fowler theorem, Centralizer and normalizer, Character table, Character theory, Conjugacy class, Group (mathematics), Group representation, Group theory, Inverse element, Involution (mathematics), Order (group theory), Real number, Symmetric group, Trace (linear algebra).
Brauer–Fowler theorem
In mathematical finite group theory, the Brauer–Fowler theorem, proved by, states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3.
See Real element and Brauer–Fowler theorem
Centralizer and normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set \operatorname_G(S) of elements of G that commute with every element of S, or equivalently, such that conjugation by g leaves each element of S fixed. Real element and centralizer and normalizer are group theory.
See Real element and Centralizer and normalizer
Character table
In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. Real element and character table are group theory.
See Real element and Character table
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix.
See Real element and Character theory
Conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b. Real element and Conjugacy class are group theory.
See Real element and Conjugacy class
Group (mathematics)
In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element. Real element and group (mathematics) are group theory.
See Real element and Group (mathematics)
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Real element and group representation are group theory.
See Real element and Group representation
Group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
See Real element and Group theory
Inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.
See Real element and Inverse element
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, for all in the domain of.
See Real element and Involution (mathematics)
Order (group theory)
In mathematics, the order of a finite group is the number of its elements. Real element and order (group theory) are group theory.
See Real element and Order (group theory)
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Real element and Real number
Symmetric group
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.
See Real element and Symmetric group
Trace (linear algebra)
In linear algebra, the trace of a square matrix, denoted, is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of.
See Real element and Trace (linear algebra)
References
[1] https://en.wikipedia.org/wiki/Real_element
Also known as Extended centralizer, Strongly real element.