Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the p-groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

0.1. For any finite group Γ, a “nonabelian Fourier transform matrix” was introduced in [L1]. This is a square matrix whose rows and columns are indexed by pairs formed by an element of Γ and an irreducible representation of the centralizer of that element (both defined up to conjugation). As shown in [L2], this matrix, which is unitary with square 1, enters (for suitable Γ) in the character for...

For a given prime p, what is the smallest integer n such that there exists a group of order p in which the set of commutators does not form a subgroup? In this paper we show that n = 6 for any odd prime and n = 7 for p = 2.

Rational automorphisms of products of simple p-compact groups are shown to be composites of products of rational automorphisms of the individual factors and permutation maps.

We describe methods for explicit computation of Galois groups of certain tamely ramified p-extensions. In the finite case this yields a short list of candidates for the Galois group. In the infinite case it produces a family or few families of likely candidates.

In this paper, we provide a survey (both theoretical and computational) of the conditions for determining which finite p -groups are strongly flat. Let m(G) denote the maximum size of an irredundant generating set of a group G . Then, a group G is strongly flat if m(G) is strictly greater than m(H) for all proper subgroups H of G . For example, J. Whiston has shown in his dissertation that all ...