Truncated versions of Dwork's lemma for exponentials
of power series and p-divisibility of arithmetic functions
(34 pages)
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.
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