Generating Alternating Groups Is there a way to think about how to generate alternating groups? Say I wanted to generate the alternating groups $A_3,A_4,A_5$.
 A: In general you can generate the alternating groups using 3-cycles.
So $A_3 = \langle(123)\rangle$
$A_4 = \langle(123), (234)\rangle$
$A_5 = \langle(123), (234), (345)\rangle$
So on and so forth.
A: Transpositions generate $S_n$, this is a well-known fact. We also know that $A_n$ is the kernel of the sign map, i.e. when we write an element of $S_n$ as a product of transpositions, the number of such transpositions that appear $\pmod 2$ is an invariant of an element of $S_n$, and those whose number of transpositions is even precisely form the elements of $A_n$. (The sign map takes a permutation to $(-1)^{\# transpositions}$, hence is an homomorphism from $S_n$ to $(\{-1,1\}, \times)$.)
Therefore the permutations of the form $(ab)(cd)$ generate $A_n$. You probably don't need all values of $(a,b,c,d) \in \{1,\cdots,n\}^4$, but the subset you choose to take nice generators depends on the usage you want to make of them. Also note there will be a lot of duplicates if you take all possible $4$-tuples, as shows the example of $A_3 = \langle (123) \rangle$.
Hope that helps,
A: For a generating set of size $2$, you can use $(1,2,3)$, together with $(1,2,3,\ldots,n)$ (if $n$ is odd) or $(2,3,\ldots,n)$ (if $n$ is even).
