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To solve the problem of obtaining all the proper subgroups of the permutation group \( S_3 \) and their cosets, we first need to understand the structure of \( S_3 \). ### Step 1: Understand the Group \( S_3 \) The symmetric group \( S_3 \) consists of all permutations of three elements. It has \( 3! = 6 \) elements, which can be listed as: - \( e \): The identity permutation. - \( p_1 = (1\ 2) \): Swaps the first and second elements. - \( p_2 = (1\ 3) \): Swaps the first and third elements. - \( p_3 = (2\ 3) \): Swaps the second and third elements. - \( p_4 = (1\ 2\ 3) \): Cycles the elements 1 → 2 → 3 → 1. - \( p_5 = __________ ___ __________ _________ ___ ______ ____ ______ _______ __________ ____ ___.
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