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  1. MHB Apprentice

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    #1
    Consider the subset of $S_4$ defined by

    $$K_4=\{(1)(2)(3)(4),(12)(34),(13)(24),(14)(23)\}$$

    Show that for all $f \in K_4$ and all $h \in S_4$, we have $h^{-1}fh \in K_4$

    I showed all the possible cycle shapes of h and am trying to show that $h^{-1}fh$ must always have cycle shape $(2,2)$, excluding the case of identity permutation.

    Just don't know where to go from here
    Last edited by Euge; November 23rd, 2016 at 14:09. Reason: improve formatting

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    #2
    Hi, Confusedalways! Welcome.

    You're on the right track. Consider the following fact. If $\sigma = (a_1 \cdots a_r)$ is a cycle in $S_n$ and $\tau\in S_n$, then $\tau \sigma \tau^{-1} = (\tau(a_1)\cdots \tau(a_r))$. Take for instance $(12)(34)$. For all $h\in S_4$, $$h(12)(34)h^{-1} = h(12)h^{-1}h(34)h^{-1} = (h(1)\;h(2))\,(h(3)\;h(4))$$

    so $h(12)(34)h^{-1}$ has cycle structure $(2,2)$. The same argument applies to the others.

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