Geodesic
Random generation with cycle type restrictions
Eberhard, Sean1 ; Garzoni, Daniele2
1 Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB, U.K.
2 Dipartimento di Matematica “Tullio Levi–Civita” Università degli Studi di Padova Padova, Italy
Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 1-25.

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We study random generation in the symmetric group when cycle type restrictions are imposed. Given π,π′∈Sn, we prove that π and a random conjugate of π′ are likely to generate at least An provided only that π and π′ have not too many fixed points and not too many 2-cycles. As an application, we investigate the following question: For which positive integers m should we expect two random elements of order m to generate An? Among other things, we give a positive answer for any m having any divisor d in the range 3≤d≤o(n1/2).

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DOI : 10.5802/alco.149
Classification : 20B30, 20P05
Mots-clés : symmetric group, random generation.

Eberhard, Sean 1 ; Garzoni, Daniele 2

1 Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB, U.K.
2 Dipartimento di Matematica “Tullio Levi–Civita” Università degli Studi di Padova Padova, Italy
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Eberhard, Sean; Garzoni, Daniele. Random generation with cycle type restrictions. Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 1-25. doi : 10.5802/alco.149. https://geodesic-test.mathdoc.fr/articles/10.5802/alco.149/

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