Enumerating finite groups and their defining relations II

Mann, Avinoam

doi:10.1016/j.jalgebra.2004.02.003

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Corollary1

The Number Of Finite Groups With a Bounded Number Of Definin1

Introduction1

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Cited by 2 publications

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“…We believe that our counting results will have a number of such applications. As an illustration in Section 13 we confirm a conjecture of Mann [36] by a probabilistic argument Theorem 10. There exists a constant c such that the number of groups of order n that can be defined by r relations is at most n cr .…”

Section: Corollarysupporting

confidence: 73%

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Random generation of finite and profinite groups and group enumeration

Jaikin–Zapirain

Pyber

2011

Ann. Math.

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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 transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal.As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most n cr as conjectured by Mann.

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Section: Corollarysupporting

confidence: 73%

“…In [36] this bound was established for the number of nilpotent groups of order n that can be defined by r relations. In this section we prove the conjecture as a corollary to our results on irreducible representations of finite groups.…”

Section: The Number Of Finite Groups With a Bounded Number Of Defininmentioning

confidence: 99%

Random generation of finite and profinite groups and group enumeration

Jaikin–Zapirain

Pyber

2011

Ann. Math.

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“…and , groups of are also well (1993) showed isomorphic groups of order n * + !" , , for all is the order of the group, minimum number of Avinoam Mann (2006) ber of non-isomorphic , (where … which generated by k !" !"…”

Section: Introductionmentioning

confidence: 99%