Cohen and Lenstra have given a heuristic which, for a fixed odd prime p, leads to many interesting predictions about the distribution of p-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field K, the Galois group of the p-class tower of K, i.e.

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete d × N matrix is finitely rank-r completable if there are at most finitely many rank-r matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a characterization of finitely completable observation sets. We use this characterization to derive sufficient deterministic sampling conditions for unique completability. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if O(max{r, log d}) entries per column are observed.

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.

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Much is known regarding images of p-adic Galois representations coming from subquotients ofétale cohomology groups of varieties over number fields. In particular, the Mumford-Tate conjecture gives them up to subgroups of finite index, and has been proved in many cases by Serre and others. In the analogous situation of arboreal Galois representations little is known. In this paper we give a conjectural description of their images in the case that they arise via iteration of a given polynomial. We also discuss in depth the case of iteration of a quadratic polynomial with integer coefficients.