Originally Posted by

**moocow** Hello. I recently had an Algebra exam that had a few questions about graph automorphisms. I'm not too familiar with graphs (a little bit familiar with permutations, groups, and morphisms though) so I was hoping someone could help me out with these questions and/or maybe lead me to any online resources about the subject since my textbook doesn't say much about it.

1) Let $\displaystyle T$ be the graph on 4 vertices with edges {1,2} and {3,4} (union of two disjoint edges). Make a list of all elements in the automorphism group of $\displaystyle T$

This one seems easiest, so if someone could explain how to go about this maybe I would understand the concept a bit better and try figuring out the other two.

2) Let $\displaystyle T_n$ be the union of *n* disjoint edges (on 2n vertices) and $\displaystyle G_n = Aut(T_n)$. Determine the order $\displaystyle |G_n|$

I think the answer to this one was $\displaystyle n!*2^n$ but I'm not sure how to get at that.

3) (Follow up to 2) Let $\displaystyle G=G_3$ in the preceding problem. Describe all the elements of order 3 in $\displaystyle G_3$. How many are there?

Again on this one I know the answer is 8 elements of order 3, but have no idea how to find that. The elements are listed as permutations and I know the elements of order three are products of disjoint 3 cycles, but I don't know how to find them.

I'd greatly appreciate any help, since I'm not really familiar with graphs at all.