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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- Jun 1st 2009, 08:01 PMyvonnehr[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. - Jun 1st 2009, 09:58 PMGamma
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 $