We consider the probability p(Sn) that a pair of random permutations generates either the alternating group An or the symmetric group Sn. Dixon (1969) proved that p(Sn) approaches 1 as n→∞ and conjectured that p(Sn) = 1 − 1/n+o(1/n). This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an elementary proof of this result; specifically we show that p(Sn) = 1 − 1/n+O(n−2+ε). Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012).
|Number of pages||16|
|Publication status||Published - 01 Apr 2019|
Bibliographical notePublisher Copyright:
© 2018, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics