Symmetric Group

• The Symmetric Group of order is the group whose set is the set of all permutations of elements and whose binary operation is the composition of permutations. It is denoted as .

• A Permutation Group is a group whose set consists of permutations and whose binary operation involves composing two permutations. It is a Subgroup of .

• (Fraleigh 8.16) Cayley’s Theorem: Every group is isomorphic to a permutation group.

  • Intuition: Every element of the group induces a group action on the group which permutes the group elements. A mapping from group elements to permutations of the group elements gives us an isomorphism to a permutation group.

  • We can define two maps. Let be the permutation on the group defined as

    The map defined by is the left regular representation of .

    Similarly, let be the permutation on the group defined as .

    The map defined by is the right regular representation of .

  • We can think of the left and right regular representations as encoding group actions and defining a bijection from a group element to its corresponding group action.

• Let be a set. Then is transitive on if such that .

• (Fraleigh e9.29) Every subgroup for , either all permutations in are even or exactly half of them are even.

  • Intuition: Any odd permutation has a corresponding group action which maps the group to itself. In , the evens become odds and the odds become evens but the number of elements must stay the same because the group elements did not change.

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh