Permutations and Cyclic Groups

Suppose $$S$$ is a finite set having $$n$$ distinct elements. Then a one-one mapping of $$S$$ onto itself is called… Click here to read more

Two permutations $$f$$ and $$g$$ of degree $$n$$ are said to be equal if we have $$f\left( a \right) =… Click here to read more

If $$I$$ is a permutation of degree $$n$$ such that $$I$$ replaces each element by the element itself, $$I$$ is… Click here to read more

The products or composite of two permutations $$f$$ and $$g$$ of degree $$n$$ denoted by $$fg$$ is obtained by first… Click here to read more

If $$f$$ is a permutation of degree $$n$$, defined on a finite set $$S$$ consisting of $$n$$ distinct elements, by… Click here to read more

Let $$f$$ be a permutation on a set $$S$$. If a relation $$ \sim $$ is defined on $$S$$ such… Click here to read more

A permutation of the type \[\left( {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}& \cdots &{{a_{n – 1}}}&{{a_n}} \\ {{a_2}}&{{a_3}}&{{a_4}}& \cdots &{{a_n}}&{{a_1}} \end{array}} \right)\] is called… Click here to read more

Theorem 1: The product of disjoint cycles is commutative. Proof: Let $$f$$ and $$g$$ be any two disjoint cycles, i.e…. Click here to read more

The set $${P_n}$$ of all permutations on $$n$$ symbols is a finite group of order $$n{!}$$ with respect to the… Click here to read more

A permutation is said to be an even permutation if it can be expressed as a product of an even… Click here to read more

Suppose $$G$$ is a group and the composition has been denoted by multiplicatively, let $$a \in G$$. Then by closure… Click here to read more

If $$G$$ is a group and $$a$$ is an element of group $$G$$, the order (or period) of $$a$$ is… Click here to read more

Theorem 1: The order of every element of $$a$$ finite group is finite. Proof: Let $$G$$ be a finite group… Click here to read more

A group $$G$$ is called a cyclic group if, for some $$a \in G$$, every element $$x \in G$$ is… Click here to read more

Theorem 1: Every cyclic group is abelian. Proof: Let $$a$$ be a generator of a cyclic group $$G$$ and let… Click here to read more