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Nearly all the books I read give $S_n$ $(n \geq 2)$ and the generating set $\{(i,i+1) | 1 \leq i < n \}$ as an example when talking about presentation groups. But is $\{(i,i+1) | 1 \leq i < n \}$ the least set of generators, i.e., is the order of any generating set for $S_n$ equal to or greater than $n-1$? If it is the least, how to prove? Are there any other least set of generators? In general, what do these least sets look like?

Forgive me for so many questions. Thanks sincerely for any answers or hints.

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    See also http://math.stackexchange.com/questions/143999/generators-for-s-n2012-05-23

2 Answers 2

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I think $(1,2), (1,2,\ldots, n)$ is also a set of generators.

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    Thanks for everyone. In fact, if $(1,2,\cdots, n)$ is denoted as $a$, then $a (1,2) a^{-1} = (2,3)$, $\cdots$, $a (n-2, n-1) a^{-1} =(n-1,n)$. Thus, the generating set \{(i,i+1) | 1 \leq i I referred is obtained.2011-11-28
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It is proved in

J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205.

that the probability that a random pair elements of $S_n$ generate $S_n$ approaches $3/4$ has $n \to \infty$, and the probability that they generate $A_n$ approaches $1/4$.

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    The article by Isaacs and Zieschang is "Generating Symmetric Groups" in Amer. Math. Monthly 102 (1995), 734-739. A link to it online is http://www.fmf.uni-lj.si/~potocnik/poucevanje/SeminarI2013/T13-JordanovIzrek.pdf2015-01-03