of permutation groups.

I'm a little confused as what a question is asking... working in $\displaystyle S_5$, it says to find a subgroup H that is generated by (123), (23), and (45) (written in cycle notation). Not sure what that means... from what I've learned of cycle notation, i thought (for example) the first would be:

1->1

2->2

3->3

4->4

5->5

and the second:

1->2

2->3

3->??

4->4

5->5

and the third:

1->4

2->5

3->3

4->1

5->2

but then i don't know how to 'generate' a group with these. my understanding is that a generator a produces elements of the subgroup $\displaystyle a^2,a^3,...,a^n$... but i don't know how to multiply these by themselves to produce new elements.