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    Symmetric and Galois group

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    Let S_3 be the symmetric group on the set {1,2,3}. Show that S_3 is solvable and that the Galois group Gal(P/Q) of the polynomial P=Y^3+pY+q over Q is a subgroup of S_3. (See attached)

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    Solution Preview

    Notice that [S_3, S_3], the commutator subgroup of S_3, contains (123) = [(12), (13)],
    so contains <(123)>, hence has order 3 or 6, by Lagrange.

    Observe that any commutator is either equal to the identity
    or a 3-cycle. Indeed, if s, t are transpositions, then either [s, t] = 1, the identity, or a 3-cycle;
    if s, t are both 3-cycles, then either [s, t] = 1 or a 3-cycle. If s is a transposition, and t is a 3-cycle, then
    [s, t] = 3-cycle.

    Since every element of [S_3, S_3] is a product ...

    Solution Summary

    This solution provides an example of proving a group is solvable.