Q:

Can someone tell me how can I generally approach problems involving subsets forming subgroups?

Suppose, I am given two elements and asked to show that they generate a given subgroup of a group, how can I show that ? I have read about the closures but don't know how to apply them in problems.
 For example, I found two questions, one asks to show that $(1234)$ and $(1243) $ generates $S_4$, and the other one was involving matrices where I have to show that two $2\times 2$ matrices one having the $ a_12 $ position $0 $ and the others $1$, while the other one has the $ a_21 $ position $0$ and the others $1$, they generate special linear group of $2\times 2 $ matrices over $\mathbb{Z}_3$ which is a subgroup of the whole group $G$ of $2\times 2$ matrices. 
Thank you in advance .

A:

For the symmetric group $S_n$ there are several sets of generators known, e.g. $(12\cdots n)$ and $(12)$, or many others. The proofs give some insight how to deal with this questions. Keith Conrad has a note on Generating Sets, where several such results are given (see below).
Reference: Generators of $S_n$
Conrad's notes here; Example $1.8$, page $2$.

