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Math Help - Permutation representation argument validity

  1. #1
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    Permutation representation argument validity

    Hello,
    I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you.


    Problem statement: Let G be a group of order 150. Let H be a subgroup of G of order 25. Consider the action of G on G/H by left multiplication: g*aH=gaH. Use the permutation representation of the action to show that G is not simple.

    My attempt: Let S_6 be the group of permutations on G/H. Then, the action of G on G/H defines a homomorphism f:G \rightarrow S_6. We know the order of S_6 is 720. Since the order of G does not divide 720, and f(G) is a subgroup of S_6, f cannot be one-to-one. Thus, $\exists$ $g_1,g_2$ distinct in $G$ such that $f(g_1)=f(g_2) \implies f(g_1g_2^{-1})=e$. Thus, $\ker(f) = \{g:f(g)=e\}$. Since $\ker(f)$ is a normal subgroup of $G$, we have found a normal subgroup of $G$. Also, since $f$ is non-trivial, then $\ker(f)$ is a proper normal subgroup of $G.$ Hence $G$ is not simple.

    Any suggestions or corrections?
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  2. #2
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    Re: Permutation representation argument validity

    I would add a reason why the action of G on G / H defines a homomorphism f: G \to S_6. Namely, [G:G/H] = 6.
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  3. #3
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    Re: Permutation representation argument validity

    Hi,
    Unless you have previously discussed the permutation representation of a group G on the left cosets of a subgroup H of G, I would provide more detail on why any g in G induces a permutation (one to one onto mapping) by left multiplication of said cosets. Finally, the symbol G/H is normally reserved for the quotient group (so H must be normal in G), and not the left cosets of H in G. Otherwise your proof is fine.

    Your statement follows immediately from the proposition:

    Let G be a finite group with a subgroup H of index n. Then G has a normal subgroup N with n dividing |G/N| and |G/N| divides n!.

    You might want to try and prove this. Hint: G acts transitively on the left cosets of H.
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