Presentations for alternating groups Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators 
$a_n=(123),b_n=(1,2,\ldots,n)$?
 A: 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. 
A: There are few known presentations for $A_n$. You can express their generators in $a_n=(123)$, $b_n=(1,2,\ldots,n)$ and obtain some presentations in these generators. For example, take the Carmichael's presentation for $A_n$ (for any $n$) from p.172 of:
Carmichael R.D. Introduction to the theory of groups of finite order. Boston. 1937. 
Generators: $V_i=(1,2,i+2)$, $i=1,...,n-2$.
Relators: $V_1^3=V_2^3=\dots=V_{n-2}^3=1$, $(V_iV_j)^2=1$ for $1\le i<j\le n-2$.
Now express $3$-cycles $(1,2,i)$ inductively in your generators $a_n$, $b_n$ as follows: $(1,2,3)=a_n$, and
$$
(1,2,i+1)=a_nb_n^{-1}(1,2,i)b_na_n^2
$$
(we multiply permutations left-to-right, as if they are acting on the right). This way you get some presentation in your generators. It may be not the shortest one, but it will work.
