Explore BrainMass

Explore BrainMass

    Conjugacy classes

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    (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.

    © BrainMass Inc. brainmass.com March 4, 2021, 9:13 pm ad1c9bdddf

    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.