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    Conjugacy classes

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    (a) Write out the conjugacy classes explicitly in S_3 and S_4.

    (b) What are the conjugacy classes in A_4?

    (c) Since |A_4|=12, any subgroup of order 6 would be normal. Use (b) to show that A_4 has no subgroup of 6. Conclude that the converse to LaGrange's theorem is false.

    (d) Find a normal subgroup of order 4 in A_4.

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    Solution Preview

    (a) The conjugacy classes of S_3 are

    {(12), (23), (13)}
    {(123), (132)}

    Since S_3 has trivial centre (and there's a singleton conjugacy class corresponding to each element of the centre); also, conjugate elements have the same order.

    Recall that, in S_n, any two permutations having the same cycle type (that is, containing the same number of
    cycles in ...

    Solution Summary

    The solution provides an example of working with conjugacy classes and subgroups.