Search results
Results from the WOW.Com Content Network
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).
The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index . Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group 's action .
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.
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.
The main example is the ring of symmetric polynomials: symmetric polynomials are polynomials that are invariant under permutation of variable. The fundamental theorem of symmetric polynomials states that this ring is R [ σ 1 , … , σ n ] {\displaystyle R[\sigma _{1},\ldots ,\sigma _{n}]} where σ i {\displaystyle \sigma _{i}} are elementary ...
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.
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 .
Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle; Cyclic shift, also known as circular shift; Cyclic symmetry, n-fold rotational symmetry of 3-dimensional space