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Let $n\geq 5$ be odd, What is a presentation of $A_n$ with generators $a_n=(123),b_n=(1,2,\ldots,n)$?

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You need to distinguish between $n$ being odd and $n$ being even cases. In particular, $b_n=(1,2,\dots,n)$ for odd $n$ is even, and $c_n=(2,3,\dots,n)$ for even $n$ is even (how odd). Otherwise, this appears to be a difficult question: what is your motivation for asking? – Vladimir Sotirov Mar 21 '12 at 2:13
Sorry i was thinking this for odd $n$. – Balin Mar 22 '12 at 2:23

I suggest you look at 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.

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