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Truncated versions of Dwork's lemma for exponentials of power series and p-divisibility of arithmetic functions
489 - 529
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(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series with coefficients in to have coefficients in . We establish theorems on the p-adic valuation of the coefficients of the exponential of , assuming weaker conditions on the coefficients of 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.
AuthorsKrattenthaler, C; Mueller, TW
- College Publications