Group generated by two elements

[Mystic998]

Homework Statement

Essentially the problem is to show that a certain finite group (specifically the special linear group of order 2 over the finite field of 3 elements) is generated by two elements. But the problem is that it's non-Abelian, so I can't just consider powers of the generators. So I'm wondering if there's any general way to show that the generators of a group actually generate the group without going through all the possible products.

Homework Equations

I suppose the definition of a group generated by A being the intersection of all groups containing A.

The Attempt at a Solution

All my current solution attempts for the specific problems have to do with showing it by taking arbitrary products, but I just kind of stop in the middle because I know it's going to take forever if I do it that way (the group's order is 24 after all).

 

[lippyka]

I don't know if there's some really obvious way to do this that I'm missing, or if there's some kind of theorem that I could use. Any help would be appreciated.