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Math Help - Conjugancy class of a^2

  1. #1
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    Conjugancy class of a^2

    D is a subgroup of S6, D={e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b}

    I'm trying to find the conjugancy class of a^2 where a^2=(135)(246).
    I understand how u find the conjugancy class of a group like s6 but I don't understand how you would do it for a single element.
    For a group, I write down the cycle structure, number of elements and parity. So for an element do I just do the one cycle structure of (xxx)(xxx)?
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  2. #2
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    Re: Conjugancy class of a^2

    Hi,
    First, you seem a bit confused about conjugacy classes in a group. Given a group G and g an element of G, the conjugacy class of g in G is the set cl(g) = {x-1gx : x is in G}. A standard result is the number of elements in cl(g) is the index of the centralizer C(g) in G.

    Next, do you know any more about the group D? Or is that part of the problem? To find the conjugacy class of a2 in D, you need to know how products are formed in D. In particular, you need to know the element b-1ab. With the information given, (D is a subgroup of S6 and D has 12 elements), it is possible to deduce that b-1ab = a-1 and that the order of C(a2) in D is 6; i.e. the index of the centralizer is 2. So cl(a2) = {a2, a-2}.
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  3. #3
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    Re: Conjugancy class of a^2

    I think im really confused as in class when we were taught about conjugacy classes twice and they were really different things!
    I gave up with trying to do the question in the end, i think i kinda see what i have to do but im going to try and see if my lecturer will go over it with me and give some practice question.
    thanks for the help.
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