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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 S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols consists of ...
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action. Counting the square dimensions of the representations ( 1 2 + 1 2 + 2 2 = 6 {\displaystyle 1^{2}+1^{2}+2^{2}=6} , the order of the group), we see these must be all of the irreducible representations.
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.
In fact, C G (S) is always a normal subgroup of N G (S), being the kernel of the homomorphism N G (S) → Bij(S) and the group N G (S)/C G (S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as W ( G , T ) = N G ( T )/C G ( T ) , and especially if the torus is maximal (i ...
the six inequivalent structures of an abstract 6-element set as the projective line P 1 (F 5) – the line has 6 points, and the projective linear group acts 3-transitively, so fixing 3 of the points, there are 3! = 6 different ways to arrange the remaining 3 points, which yields the desired alternative action.
The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1]
In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G.