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## Homework Statement

Consider the group of permutation on the set {123}. Is this group cyclic? Justify your answer

## Homework Equations

## The Attempt at a Solution

I wrote out the cayley table for this group, and noticed that if we take (123)^3 = e . Seeing as we can get back to the original orientation of the permutation by composition of (123) three times and that any permutation can be written as a product of transpositions is this enough to show that the group is cyclic? I think it is but im not totally convinced.

Also what happens if we are given a larger group and have to show that it is cyclic or not? Say {1234567}. Is there a quicker way than writing out all permutations manually trying to find some (g^n) = e (where n is a member of the integers) ?

Thanks