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Thread: Isomorphism

  1. #1
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    Isomorphism

    Find the minimum of all positive integers m such that every sigma in S_9 (Sigma in A_9) satisfies sigma^ n = 1.

    Find 4 different subgroups of S_4, isomorphic to S_3, and nine isomorphic to S_2. ( I dont need all the subgroups just how to find them isomorphic)

    Let G be a subgroup of S_5. Prove that if G contains a 5-Cycle and a 2-Cycle, then G = S_5.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Juancd08 View Post
    Find the minimum of all positive integers $\displaystyle m$ such that every $\displaystyle \sigma \in S_9$ ($\displaystyle \sigma \in A_9$) satisfies $\displaystyle \sigma^ n = 1$.
    Hint: for $\displaystyle \sigma = (a_1 a_2 \ldots a_i)$ a permutation of $\displaystyle i$ objects then $\displaystyle \sigma$ has order precisely $\displaystyle i$. Prove this, then apply it to your questions.

    Quote Originally Posted by Juancd08 View Post
    Find 4 different subgroups of $\displaystyle S_4$, isomorphic to $\displaystyle S_3$, and nine isomorphic to $\displaystyle S_2$.
    To find the subgroups isomorphic to $\displaystyle S_2$ note that $\displaystyle S_2 \equiv C_2 (=\mathbb{Z}_2)$, the cyclic group of 2 elements. So, how many elements in this group have order 2?...

    For $\displaystyle S_3$, your groups are just the permutations of 3 elements, $\displaystyle \{a_1, a_2, a_3\} \subset \{1,2,3,4\}$. This is because $\displaystyle S_3$ is merely the group of all possible permutations of 3 objects, by it's "rawest" definition. If you take all possible permutations of 3 objects you will get a group isomorphic to $\displaystyle S_3$.

    Quote Originally Posted by Juancd08 View Post
    Let G be a subgroup of S_5. Prove that if G contains a 5-Cycle and a 2-Cycle, then G = S_5.
    I suspect someone here can provide you with a hint towards a much neater proof than I have. Sorry. However, I would advise you to try this for smaller values of $\displaystyle n$ to see how it works. Note that $\displaystyle S_n$ is generated by a transposition and an $\displaystyle n$-cycle for all $\displaystyle n > 2$
    Last edited by Swlabr; Jun 28th 2009 at 11:27 PM.
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