Permutations and Cyclic Groups

  • Permutations

    Suppose is a finite set having distinct elements. Then a one-one mapping of onto itself is called a permutation of degree . The number of elements in the finite set is known as the degree of permutation. Symbol for a Permutation: Let be a finite set having distinct elements. If in one-one mapping, then is […]

  • Equality of Two Permutations

    Two permutations and of degree are said to be equal if we have , . Example: If and are two permutations of degree 4, then we have. Here we see that both and replace 1 by 2, 2 by 3, 3 by 4, and 4 by 1. If is a permutation of degree , we […]

  • Identity Permutation

    If is a permutation of degree such that replaces each element by the element itself, is called the identity permutation of degree . Thus Or Or                    is the identity permutation of degree . For example, is the identity permutation on

  • Product or Composite of Two Permutations

    The products or composite of two permutations and of degree denoted by , is obtained by first carrying out the operation defined by then by . Let us suppose is the set of all permutations of degree . Let and be two elements of . Hence the permutation has been written in such a way […]

  • Inverse of Permutations

    If be a permutation of degree , defined on a finite set consisting of distinct elements, by definition is a one-one mapping of onto itself. Since is one-one onto, it is invertible. Let be the inverse of map then will also be one-one map of onto itself. Thus, is also a permutation of degree on […]

  • Orbit of Permutations

    Let be a permutation on a set . If a relation is defined on such that For some integral , we observe that the relation is (i) Reflexive, the relation is reflexive i.e. , now we can define reflexive property according to the above definition, because             (ii) Symmetric, the relation is symmetric i.e. , […]

  • Cyclic Permutations

    A permutation of the type is called a cyclic permutation or a cycle. It is usually denoted by the symbol . Thus if is a permutation of degree non a set having distinct elements and if it is possible to arrange some of the elements (say in number) of the set in a row such […]

  • Theorems of Cyclic Permutations

    Theorem 1: The product of disjoint cycles is commutative. Proof: Let and be any two disjoint cycles, i.e. there is no element common in two when they are expressed in one row notation. Therefore, the elements permuted by are invariant under and the elements permuted by are invariant under . Hence the product of disjoint […]

  • Group of Permutations

    The set of all permutations on symbols is a finite group of order with respect to composite of mappings as the operation. For , this group is abelian and for it is always non-abelian. Let be a finite set having distinct elements. Thus there are permutations possible on . If denotes the set of all […]

  • Even and Odd Permutations

    A permutation is said to be an even permutation if it can be expressed as a product of an even number of transpositions, otherwise, it is said to be an odd permutation i.e. it has odd number transpositions. Theorem 1: A permutation cannot be both even and odd, i.e. if a permutation is expected as […]

  • Integral Powers of an Element of a Group

    Suppose is a group and the composition has been denoted by multiplicatively, let . Then by closure property etc. are all elements of . Since the composition in obeys general associative law, therefore to factors is dependent of the manner in which the factors may be grouped. If is positive integer, we define factors to […]

  • Order of an Element of a Group

    If is a group and is an element of group , the order (or period) of is the least positive integer such that If there exist no such integer, we say that is a finite order or zero order. We shall use the notation for the order of . Note that the only element of […]