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Math Help - What is the square of a cycle in S_7 symmetric group?

  1. #1
    Senior Member x3bnm's Avatar
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    What is the square of a cycle in S_7 symmetric group?

    Suppose there's a cycle \alpha = (3714) in S_7 where S_7 is a symmetric group on a set \{1, 2, \cdots, 7\}

    What is \alpha^2?

    Is it composition of \alpha like this \alpha \circ \alpha?

    I'm asking this because suppose there's a function f(x) = x + 1

    Now, (f(x))^2 = (x + 1)^2 whereas (f \circ f)(x) = (x + 1) + 1

    which are completely two different things.

    So what is the meaning of \alpha^2(square of a cycle) in abstract algebra?
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  2. #2
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    Re: What is the square of a cycle in S_7 symmetric group?

    yes, "what a square is" depends on "what your multiplication is".

    with real-valued functions, there are indeed "two different multiplications" we might use: composition, or "point-wise multiplication" (the (x+1)2 one).

    however, in permutation groups, the group operation is composition (permutations are nothing more than bjiective functions on finite sets), so α2 = αoα

    so it makes sense, for example, to talk of α2(4) = α(α(4)).

    in a general group, "we don't know" what our "multiplication" is (although we do know for SPECIFIC groups). it's just a binary operation that satisfies some axioms. the word "multiplication" is sort of misleading, "group operation" is more accurate.

    in your example, (3 7 1 4)2 is this function:

    1-->4-->3
    2-->2-->2
    3-->7-->1
    4-->3-->7
    5-->5-->5
    6-->6-->6
    7-->1-->4

    which is the permutation (1 3)(4 7) (it is customary to write (3 7 1 4) as (1 4 3 7), but not necessary).

    there is a nifty trick for computing powers of a k-cycle:

    for the square, "skip one space". so instead of sending 3 to 7, we skip one and send 3 to 1. we next do the same thing with 1.
    for the cube, "skip two spaces".
    for the n-th power "skip n-1 spaces".

    so (1 4 3 7)3 sends 1 to 7 (1 7....

    and then sends 7 to (skip 1, skip 4) uh..."3"! (1 7 3....)

    and sends 3 to (skip 7, skip 1) 4, and sends 4 to (skip 3, skip 7) 1 (close cycle) (1 7 3 4).
    Last edited by Deveno; December 16th 2012 at 09:52 PM.
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  3. #3
    Senior Member x3bnm's Avatar
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    Re: What is the square of a cycle in S_7 symmetric group?

    Thanks Deveno.
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