Groups of Permutations

    One-to-one and onto mappings from a set to itself are called permutations and the collection of all permutations on a set is a group (called a permutation group) under the operation of composition of mappings. In this topic, many examples are given to explain the importance of permutation groups when the underlying set is a finite set of counting numbers; and the matrix form and cycle notation of such permutations are detailed so as to fully explore the groups of permutations of finite sets of counting numbers (called symmetric groups). But it is important to realize that permutation groups, in general, can be collections of permutations of any type of set.
    A permutation of a non-empty set is a bijective mapping from to itself. The set of all permutations on is denoted by Under composition of mappings, is a group because compositions of bijective mappings are bijective, composition is an associative operation, bijective mappings are invertible, and the identity mapping is This group is sometimes denoted by

Definition (Groups of Permutations) In general, any group whose elements are permutations, with composition as the operation, is a permutation group. When then the group is commonly denoted by  

Example (Symmetric Group of Order 6) Write out the Cayley table for the symmetric group and using matrix notation



and using cyclic notation



Can you find any subgroups of

Group Theory Books

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