From: Edwin Clark Newsgroups: sci.math.research Subject: Re: Finite Groups Generated by 2 Elements Date: Tue, 3 Nov 1998 01:19:56 -0500 Searching around on MathSciNet I finally found that it is a theorem that finite simple non-abelian groups can be generated by two elements -- one of which is an involution (element of order 2). I copy the review below. Apparently the proof depends on the classification theorem for finite simple groups! --Edwin Clark > ------------------------------------------------------------------------ > 86m:20060 20J06 (20C15 20D08) > Aschbacher, M.(1-CAIT); Guralnick, R.(1-SCA) > Some applications of the first cohomology group. > J. Algebra 90 (1984), no. 2, 446--460. > ------------------------------------------------------------------------ > The object of the article is \$H\sp 1(G,V)\$, the first cohomology group of a finite group \$G\$ and a finite \${Z}G\$-module \$V\$. It is shown that \$H\sp 1(G,V)\$ is not too big in many cases. Theorem A: If \$V\$ is a simple faithful \${Z}G\$-module, then \$\vert H\sp 1(G,V)\vert <\vert V\vert \$. The concept of the proof is as follows. Reduction to the case when \$G\$ is simple. Theorem A follows then from generation properties of simple groups. It is the main purpose of the article to establish suitable (in particular with respect to Chevalley groups) generation properties. Among other results, it is shown that the sporadic simple groups are generated by an involution and another element. This is not only relevant for Theorem A. By the classification of the finite simple groups this ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ completes the proof of the following well-known conjecture. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Theorem B: Every finite simple group can be generated by two elements. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The bound on \$H\sp 1\$ has several applications, e.g., it follows that minimal relation modules of finite simple groups are unique [cf. J. Williams and the reviewer, Arch. Math. (Basel) 42 (1984), no. 3, 214--223; MR 85k:20155]. Another application is a computation of the minimum number of generators of a finite group \$G\$ with an abelian minimal normal subgroup \$A\$ by the knowledge of the minimum number of generators of \$G/A\$. Finally, the authors give reasons for their conjecture that the number of irreducible characters of \$G\$ bounds the number of conjugacy classes of maximal subgroups of \$G\$. ============================================================================== From: rbfeinb@ccr-p.ida.org (Robert B. Feinberg) Newsgroups: sci.math.research Subject: Finite Groups Generated by 2 Elements Date: 3 Nov 1998 12:41:21 -0500 In 1972 Courant Institute of Mathematical Sciences published lecture notes by Constance Davis titled A bibliographical survey of groups with two generators and their relations, MR 54 #410. Regrettably that department no longer distributes notes, and the book hasn't been published. Consequently, I'll loan my copy to Prof. Clark if he can't obtain it otherwise. I used MathSciNet to search on the keywords "finite group" and "two" and "(generate or generators.)" The result was 370 references. No. 88 looked interesting: a Russian article by S. Strunkov in Mat. Zametki 48 (1990.) ============================================================================== From: mareg@lily.csv.warwick.ac.uk (Dr D F Holt) Newsgroups: sci.math.research Subject: Re: Finite Groups Generated by 2 Elements Date: 4 Nov 1998 12:04:27 GMT In article , Edwin Clark writes: >Searching around on MathSciNet I finally found >that it is a theorem that finite simple non-abelian >groups can be generated by two elements -- one of >which is an involution (element of order 2). I copy >the review below. Apparently the proof depends on >the classification theorem for finite simple groups! > >--Edwin Clark You may also be interested in the curious fact (first calculated by Philip Hall) that Alt(5)^19 (i.e. the direct product of 19 copies of the alternating group of degree 5) can be generated by two elements, but Alt(5)^20 cannot. For larger simple groups, you get correspondingly larger numbers. For example, let n be maximal such that PSL(2,q)^n can be generated by two elements. A couple of years ago I did some calculations with a colleague Murray Macbeath, and we got: q = 2 3 5 7 8 9 11 13 16 17 19 23 25 27 29 31 32 n = 3 4 19 57 142 53 254 495 939 1132 1570 2881 1822 1572 5825 7135 6330 (We had a conjecture for a general 'formula', which we did not quite succeed in proving.) Derek Holt. ============================================================================== From: mann@vms.huji.ac.il (Avinoam Mann) Newsgroups: sci.math.research Subject: Re: Finite Groups Generated by 2 Elements Date: 5 Nov 1998 02:41 IDT > Edwin Clark writes: >>Searching around on MathSciNet I finally found >>that it is a theorem that finite simple non-abelian >>groups can be generated by two elements -- one of >>which is an involution (element of order 2). I copy >>the review below. Apparently the proof depends on >>the classification theorem for finite simple groups! IIRC, the same result, which was a well known conjecture, was proved at about the same time by M.Cartwright. Much more is true. Netto has conjectured already last century that most pairs of elements of an alternating group generate the full group. This was proved by J.D.Dixon, who conjectured that a similar result holds for all finite simple groups. I.e. if S is a simple group of order n, and if k pairs of elements generate S, then the ratio k/n(squared) tends to 1 as n tends to infinity. For the classical groups, and small exceptional groups, this was proved by Kantor-Lubotzky, and the proof for the other exceptional groups was given by Liebeck-Shalev. The latter also proved a similar result when one of the generators is of order 2, and several related results. Avinoam Mann ============================================================================== From: Shai Dekel Newsgroups: sci.math.research Subject: Re: Finite Groups Generated by 2 Elements Date: 19 Nov 1998 10:00:02 -0600 The following simple fact is stronger, less known and might interest a few people: Theorem: for each element x in Sn (An) there exists a complement y, such that = Sn (An) Except for the case of x in the conjugancy class of (1,2)(3,4) in the group S4. A nice result on group covering you get from the above is: Let Union{Gi} = Sn (An), where Gi are subgroups <> Sn (An). Then, Intersection{Gi} = 1 Shai john baez wrote: > In article <71jtft\$f16\$1@holly.csv.warwick.ac.uk>, > Dr D F Holt wrote: > > >Yes, it is known that all of the finite simple groups can generated by > >two elements. > > Has anyone tried to use this idea, together with the fact that > SL(2,Z) is generated by two elements, to explain the mysterious > relationships between finite simple groups and conformal field > theory? (You know, monstrous moonshine and all that jazz.)