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.
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.
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 $