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Thread: Problem - cycle notations

  1. #1
    Junior Member
    Nov 2008

    Problem - cycle notations

    Can anybody help with the following:

    Consider the alternating group A4.
    a) Write the elements of A4 in cycle notation.
    b) For each element write the inverse in cycle notation.
    c) Construct the Cayley table for A4.

    I know that A4 is the set of all even permutations of S4 but I'm not sure what an even permutation is. If anybody could start me off with this problem it would be a great help.

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  2. #2
    Aug 2008
    Start by writing all elements of S_4. To be more precise A_4 is a subgroup of S_4 not only a set.

    Generally if you want to find elements in S_n then consider all the partations of n, i.e., all the different methods n can be written as sum of positive integers.
    Here we have that 4 = 4, 4=3+1, 4=2+2, 4=1+1+2, 4=1+1+1+1. These are the only elements of S_4.

    SO in cycle form we write:

    Cycle number of elments of this cycle type

    (*)(*)(*)(*) 1
    (**)(**) 3
    (*)(***) 8
    (*)(*)(**) 6
    (****) 6
    It can be little bit difficult for you to find number of elements of a given cycle type because you need some combinatoirs for that, but you will get use to it.

    A permutation is even if it can be written as a product of even number of permutations.

    I hope this helps, otherwise write again.
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  3. #3
    Junior Member
    Nov 2008
    Hey man,

    Thanks for the help. I've cracked it now.

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