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Math Help - Abstract Algebra proof help

  1. #1
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    Abstract Algebra proof help

    The problem (this is from Gallian btw) states:

    Suppose that H is a subgroup of S_4 and that H contains (12) and (234). Prove that H=S_4

    The books gives a "solution" to this problem in the back, but I'm having trouble understanding it.

    "By closure, (234)(12)=(1342) belongs to H so that |H| is divisible by 3 and 4 and divides 24."

    Can someone explain the bolded part? Then maybe I can figure out the rest of what the book says...
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  2. #2
    Member HappyJoe's Avatar
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    Sure.

    Recall that in a finite group H, the order of an element g\in H divides the order of H, i.e. \text{ord}(g)\ |\ |H|.

    The cycle (234) is contained in H, and as this cycle has order 3, we have that 3 must divide |H|. Similarly, the cycle (1342) is an element of H, and so 4 must divide |H|.

    And you also know (from Lagrange's theorem) that the order of a subgroup divides the order of the group, which is why the order of H must divide the order of S_4, which is 24.
    Last edited by HappyJoe; October 17th 2010 at 12:24 PM. Reason: Forgot something.
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