Is there a simple algorithm to generate the elements of the alternating group $A_n$ in terms of some small set of generators?
For example, when $n = 4$, I'm looking for an algorithm whose output is a generating set, say $\{ a = (1,2,3), b = (1,2)(3,4) \}$ together with a list of 12 words in $a$ and $b$ that represent that elements of $A_4$.
What about such an algorithm for the symmetric group? For example, when $n = 4$ an algorithm that outputs, for example, $\{ a = (1,2,3,4), b = (1,2) \}$ and a listing of $S_4$ as 24 words in $a$ and $b$.
For the symmetric group $S_n$, you can generate it with the two elements $a=(1,2)$ and $b=(1,2,\dots,n)$: By conjugating $a$ by $b^k$, you get $(1+k,2+k)$; this allows you to swap any two adjacent elements $1,\dots,n$ using only the generators $a$ and $b$. Given an arbitrary permutation, you can then bring it to the identity permutation by using the bubble-sort algorithm; if you keep track of the steps in this process, it will show you how to express the permutation as a word in $a$ and $b$.
This technique can also be adapted to show that the alternating group $A_n$ can be generated by two elements, $a=(1,2,3)$ and $b=(1,2,\dots,n)$ (if $n$ is odd) or $b=(2,\dots,n)$ (if $n$ is even).