Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions

Abstract

(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series S(z) with coefficients in Qp to have coefficients in Zp. We establish theorems on the p-adic valuation of the coefficients of the exponential of S(z), assuming weaker conditions on the coefficients of S(z) than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the p-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian p-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo p for free products of finite Abelian p-groups, results on p-divisibility of permutation numbers with restrictions on their cycle structure, and a curious “supercongruence” for a certain binomial sum.