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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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    Sorry i was thinking this for odd $n$.2012-03-22

2 Answers 2

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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.

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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.

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.