Corrigendum: Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces

Strongly Real Beauville Groups

“…from which we have that this Beauville structure is strongly real. Finally for M 23 ×M 23 we similarly have that the permutations (1,3,19,7,18,4,11,21,16,14,6) (2,23,9,17,15,20,22,10,13,12,8) ( 24,45,39,35,34,37,25,27,32,30,38) (26,36,43,29,40,28,44,46,41,33,31) y 2 := (1,6,22,9,16,17,5,19,11,18,2) define a strongly real Beauville structure of type ((23,23,23), (11,…”

“…This lead Bauer, Catanese and Grunewald to conjecture that aside from A 5 , which is easily seen to not be a Beauville group, every non-abelian finite simple group is a Beauville group -see [6, Conjecture 1] and [7,Conjecture 7.17]. This suspicion was later proved correct [19,20,25,26], indeed the full theorem proved by the author, Magaard and Parker in [19] is actually a more general statement about quasisimple groups (recall that a group G is quasisimple if it is generated by its commutators and the quotient by its center G/Z(G) is a simple group. ).…”

On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

“…On the other hand, all finite simple groups other than A 5 are Beauville groups, as shown by Guralnick and Malle in 2012 [12]. Fairbairn, Magaard and Parker [5,6] proved more generally that all finite quasisimple groups are Beauville groups, with the only exceptions of A 5 and SL 2 (5). Additionally, Garion, Larsen and Lubotzky obtained an 'almost all' version of the aforementioned results using probabilistic methods (see [10]).…”

Coxeter groups as Beauville groups

“…x 1 := (1, 2, 3, 4, 5)(6, 7)(8, 9)(10, 11), y 1 := (1, 2)(3, 4)(8, 9, 10, 11, 12)(13, 14), x 2 := (1, 2, 3)(8, 9)(10, 11)(13, 14), y 2 := (1, 4)(2, 5)(6, 7)(8, 10, 12) (13,14) whilst the following permutations give a strongly real Beauville structure for the group W (A 4 ) × W (I 2 (3)) It is also easy to see that neither W (A 4 ) × W (I 2 (5)) nor W (A 5 ) × W (I 2 (5)) are Beauville groups since for any generating pair x and y we have that Σ(x, y) must contain elements of order 5 that belong solely in the 'I 2 part' of the group. Similarly if k is coprime to |K 1 |, then K 1 × W (I 2 (k)) cannot be a Beauville group, nor can W (I 2 (k)) × W (I 2 (k ′ )) be a Beauville group for any k and k ′ .…”