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Thread: [SOLVED] Using Cayley's Theorem

  1. #1
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    [SOLVED] Using Cayley's Theorem

    Can someone help me with the following problem?

    Let G be the cyclic group <a> of order 5.
    1. Write out the elements of a group of permutations that is isomorphic to G.
    2. Exhibit an isomorphism from G to this group.

    Thanks.
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  2. #2
    Super Member Gamma's Avatar
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    Well you know it should be isomorphic to a subgroup of $\displaystyle S_5$ and it needs to be cyclic, so you need only find an element of $\displaystyle S_5$ which has order 5. (1 2 3 4 5) is a good choice, the isomorphism is as follows:

    $\displaystyle a \rightarrow (1 2 3 4 5)$
    $\displaystyle a^2 \rightarrow (1 2 3 4 5)^2=(1 3 5 2 4)$
    $\displaystyle a^3 \rightarrow (1 2 3 4 5)^3=(1 4 2 5 3)$
    $\displaystyle a^4 \rightarrow (1 2 3 4 5)^4=(1 5 4 3 2)$
    $\displaystyle a^5=e \rightarrow (1 2 3 4 5)^5=(1)(2)(3)(4)(5)=e $
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