It is well known that a permutation group of degree $n \neq 3$ can be generated by $[\frac{n}{2}]$ elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree $n$ with $[\frac{n}{2}]$ elements. In particular we prove that if $n$ is large enough and $[\frac{n}{2}]$ elements generate a permutation group $G$ of degree $n$ modulo $G^\prime G^2,$ then almost certainly these elements generate $G$ itself.
THE PROBABILITY OF GENERATING A PERMUTATION GROUP / Lucchini, A; Morini, Fiorenza. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - 82:(2004), pp. 395-403. [10.1007/s00013-004-0808-6]
THE PROBABILITY OF GENERATING A PERMUTATION GROUP
MORINI, Fiorenza
2004-01-01
Abstract
It is well known that a permutation group of degree $n \neq 3$ can be generated by $[\frac{n}{2}]$ elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree $n$ with $[\frac{n}{2}]$ elements. In particular we prove that if $n$ is large enough and $[\frac{n}{2}]$ elements generate a permutation group $G$ of degree $n$ modulo $G^\prime G^2,$ then almost certainly these elements generate $G$ itself.| File | Dimensione | Formato | |
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