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The following polynomials in two variables X 1 and X 2 are symmetric: + + + + (+) as is the following polynomial in three variables X 1, X 2, X 3: . There are many ways to make specific symmetric polynomials in any number of variables (see the various types below).
For every symmetric group other than S 6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. Or as follows: Each permutation of order two (called an involution ) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2 k 1 n −2 k .
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the elementary symmetric polynomials are the most ...
For example, the 8-element quaternion group has cycle graph shown at right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well.
The Schur multipliers of the alternating groups A n (in the case where n is at least 4) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is the cyclic group of order 6, and the covering group is a 6-fold cover. H 2 (A n, Z) = 0 ...
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.