2011
DOI: 10.4007/annals.2011.173.2.4
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Random generation of finite and profinite groups and group enumeration

Abstract: We obtain a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice.Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a tran… Show more

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Cited by 32 publications
(64 citation statements)
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“…Remarkable results characterizing PFG profinite groups have been very recently obtained by Jaikin-Zapirain and Pyber [26]. Theorem 1 in that paper provides strong bounds on ν(G) for G finite.…”
Section: By [42 12] We Have M(g) < ν(G) + 4 For Any Finite Group Gmentioning
confidence: 76%
“…Remarkable results characterizing PFG profinite groups have been very recently obtained by Jaikin-Zapirain and Pyber [26]. Theorem 1 in that paper provides strong bounds on ν(G) for G finite.…”
Section: By [42 12] We Have M(g) < ν(G) + 4 For Any Finite Group Gmentioning
confidence: 76%
“…We find a useful way of characterising profinite groups with this property. This characterisation allows us to actually work with the APFG property and it is not difficult to prove, by using the different characterisations of PFG groups given in [17], [16] and [10], that PFG groups are APFG. It is worth mentioning that the converse does not hold, as is shown in Example 4.5; moreover, we produce an example of a finitely generated profinite group which is not APFG.…”
Section: Theorem 2: Let G Be a Countably-based Profinite Group And Asmentioning
confidence: 97%
“…As previously noted, δ(H) < 1 for all H ∈ M 1 (G), so The constant γ here can be expressed in terms of the undetermined constant β in Theorem 4.10. It would be desirable to have an effective result with an explicit constant, but it seems rather difficult to extract constants from the proof of Theorem 4.10 in [33].…”
Section: Subgroup Growthmentioning
confidence: 99%