A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p

Abstract

We explicitly construct infinitely many a non-abelian strongly real Beauville p-groups for every prime p. Until very recently, only finitely many non-abelian strongly real Beauville p-groups were known and all of these were 2-groups.

An infinite family of strongly real Beauville p-groups

“…Let us start with the case a = xy and b = w 1 . Observe that b and xy 2 lie in the same maximal subgroup of G, and this, together with (3), implies that (b g ) 3 k = (xy 2 ) 3 k , for all g ∈ G. Recall that by the proof of Theorem 2.1, for any n ∈ N we have (4) (xy)…”

“…On the other hand, if p = 3 then since γ 2 (T )/γ 3 (T ) is cyclic and γ 3 (T )/γ 4 (T ) is a 2-generator group, and both are of exponent at most 3 k , it then follows that |T /γ 4 …”

“…I thank Ben Fairbairn for providing me with his preprints [4] and [5]. Also, I wish to thank the Department of Mathematics at the University of the Basque Country for its hospitality while this paper was being written.…”

“…She constructed an infinite family of non-abelian strongly real Beauville pgroups by considering the lower central quotients of the free product of two cyclic groups of order p. However, this result does not cover the prime 2 and also does not give a strongly real Beauville p-group of every possible order. At around the same time, Fairbairn [4] gave another infinite family of non-abelian strongly real Beauville p-groups for odd p, by using wreath products of cyclic p-groups.…”

Beauville p-Groups: A Survey

“…At around the same time the author constructed another infinite family of nonabelian strongly real Beauville p-groups for p odd in [23] by proving the following.…”