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Math Help - alternating group

  1. #1
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    alternating group

    Hello!

    Can somebody help me with the following problem? Let A_5 be the alternating group of order 5.

    a) Which orders can a given element of A_5 possibly have?
    b) Let n be a possible order as in a). How many elements of order n does A_5 have?

    This seems to work with the Sylow theorems, but how can one proceed?

    Marco
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  2. #2
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    Re: alternating group

    we can answer this completely just by looking at the possible disjoint cycle types in A5:

    5-cycles (which can be written as a product of 4 transpositions: (a b c d e) = (a e)(a d)(a c)(a b))
    3-cycles (which can be written as a product of 2 transpositions: (a b c) = (a c)(a b))
    2 disjoint 2-cycles such as (a b)(c d)
    the identity (which can be written as a product of 2 transpositions: e = (a b)(a b)).

    which have orders 5,3,2 and 1, respectively.

    the second part of your question can then be answered by asking how many 5-cycles, etc. there are in S5, since for any cycle type in A5, A5 contains every cycle of that type.
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  3. #3
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    Re: alternating group

    Thank you, Deveno!

    Can this be used to deduce hat A_5 cannot have a nontrivial subgroup?

    Marco
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