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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 ...
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.
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.
If R is a ring or an algebra over a field, and S is a subset of R, then the centralizer of S is exactly as defined for groups, with R in the place of G.. If is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset S of is defined to be [4]
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] Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras over a field .
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.
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras.The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and ...
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.