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Lecture# Symmetric Group and Alternating Group

Description

This lecture covers the Symmetric group, cycle notation, multiplication, and conjugation in S_n, the sign of a permutation, and the Alternating group A_n. It explains how any element of S_n is a product of disjoint cycles, uniquely up to the order of factors. The lecture also discusses the kernel of a permutation, the alternating group as a normal subgroup of even permutations, and the cycle decomposition of permutations.

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In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n.

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