Algebra: Group Theory Question? (notation help)?

Question

My homework assignments defines a group H = <(123)>.

I read that as 'the cyclic group generated by the permutation (123)', but I don't see how one group can generate another. I know this is simple, but I can't figure it out.

Could someone list the elements, or clarify the notation? Thanks.

Answer 1

You read correctly. In this case, the group operation is function composition, so we need to find the results of applying the permutation f=(123) repeatedly.

f⁰({1, 2, 3}) = {1, 2, 3}
f¹({1, 2, 3}) = {3, 1, 2}
f²({1, 2, 3}) = {2, 3, 1}
f³({1, 2, 3}) = {1, 2, 3}

After that the cycle repeats, so the elements of the group are f⁰, f¹, and f², which are the permutations (), (123), and (132), respectively.

Answer 2

You were just having a minor brain cramp.

A permutation isn't a group. :)