Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators $a_n=(123),b_n=(1,2,\ldots,n)$?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
I suggest you look at http://www.math.auckland.ac.nz/~obrien/research/an-sn-present.pdf to get some idea of the current state of knowledge about this question. Theorem 1.3 states that $A_n$ has a 2-generator presentation with $O(\log n)$ relations and length $O((\log n)^2)$.
Typing $$\rm presentation\ alternating\ group$$ into Google got me this and other references.