A Note on Beauville p-Groups

Abstract

We examine which p-groups of order ≤ p 6 are Beauville. We completely classify them for groups of order ≤ p 4 . We also show that the proportion of 2-generated groups of order p 5 which are Beauville tends to 1 as p tends to infinity; this is not true, however, for groups of order p 6 . For each prime p we determine the smallest non-abelian Beauville p-group.

On rigid compact complex surfaces and manifolds

“…where ϕ : G → G/[G, G] satisfies ϕ(x) = (1, 0) = e 1 , ϕ(y) = (0, 1) = e 2 and ϕ(z) = ϕ(w) = ϕ(t) = 0. By [BBF12] G admits a Beauville structure of type (3, 3, 9), (3, 3, 9) given by (x, y, (xy) −1 ), (xt, y 2 w, (xty 2 w) −1 ), i.e., we get two triangle curves λ i :…”

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

“…Thus, T is either in H(n) or H(n + 1), and |F : T | = p t−sp a , where s = r or r − 1. In any case, we have s = t/2p a + O (1). o(1)).…”

“…Failing to find a general recipe to tell apart Beauville ‐groups from non‐Beauville ones, we can reformulate our goal and try to determine the asymptotic behaviour of the number of Beauville groups. This approach first appeared in , where the asymptotics of the number of Beauville groups of order and (as tends to infinity) were investigated.…”

“…In this article, we will consider the asymptotic behaviour of the number of Beauville groups of order as tends to infinity (cfr. [, Question 1.8]). In this respect, observe that Jaikin‐Zapirain provided asymptotic bounds for the number of ‐generator finite ‐groups of order , which reduce to the following in the case : Note that these are logarithmic estimates of the growth of , in the same vein as the result of Higman and Sims that the total number of finite ‐groups of order is asymptotically .…”

“…In this article, we will consider the asymptotic behaviour of the number of Beauville groups of order p t as t tends to infinity (cfr. [1,Question 1.8]). In this respect, observe that Jaikin-Zapirain [16] provided asymptotic bounds for the number f d (p t ) of d-generator finite p-groups of order p t , which reduce to the following in the case d = 2:…”