Two-generator groups. I.

Brenner, J. L.; Wiegold, James

doi:10.1307/mmj/1029001421

Cited by 35 publications

(57 citation statements)

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“…Sp(6, 2), 2 6 : A 8 , 2 6 : the associated linear group, and hence deal with linear transformations s rather than elements in the simple group. The first two columns of the tables list the dimension d (in Table 3, also the type and the decomposition of the natural module under s is listed) and the size q of the defining field (note that in the unitary case, the natural module is F d q 2 ), the third column lists the order |s| of the element s in the linear group, the fourth column lists the collection M(G, s) of maximal subgroups of G that contain s, and the last two columns list the quantities σ (G, s) and P (G, s) (the latter only if the former is not smaller than 1/3).…”

Section: Computer Calculationsmentioning

confidence: 99%

“…For Sp (6,2) and Sp (8,2), we computed that part (c) (and for the latter group also part (b)) of Proposition 5.8 hold. Table 3 lists the examples computed for orthogonal groups.…”

Section: G = Sp(2m Q)mentioning

confidence: 99%

“…. , x k ∈ G, there is some y ∈ G such that G = x i , y whenever 1 i k. There are quite a few papers concerning spread [1,[3][4][5][6][7]12,16,20]. In [16] and [20] it was shown that all but at most finitely many simple groups have spread at least 2, and there are infinitely many simple groups (such as Sp(2m, 2) for m 3) that have spread exactly 2.…”

Section: Introductionmentioning

confidence: 99%

“…Except when S is one of the groups + (8, 2), P + (8, 3), or A 6 , s can be chosen from a prescribed conjugacy class of elements in S (independent of G). In the first two of these exceptional cases, s can be chosen from a prescribed Aut(S)-conjugacy class of elements in S. For S = A 6 , computation shows that no such restriction is possible.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic generation of finite simple groups, II

2008

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In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 = x ∈ G, the probability is greater than 1/10 that G = x, y for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds.We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x 1 , x 2 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , y . Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x 1 , x 2 , x 3 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , y = x 3 , y . We also prove analogous but weaker results for almost simple groups.

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Section: Computer Calculationsmentioning

confidence: 99%

“…For Sp (6,2) and Sp (8,2), we computed that part (c) (and for the latter group also part (b)) of Proposition 5.8 hold. Table 3 lists the examples computed for orthogonal groups.…”

Section: G = Sp(2m Q)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probabilistic generation of finite simple groups, II

2008

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“…Its subgroups can be found in [16,17,30]; each is either soluble of derived length at most 3 or of the form PSL(2, D), 2 G 2 (D) or η × PSL(2, D), for some subfield D of F and some involution η. Since PSL(2, D) can be generated by two elements one of which has odd order (see, for instance, [4]), it follows easily that 2 G 2 (K) ∈ S 3 (2).…”

Section: Propositionmentioning

confidence: 99%