# 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 **Q**_{p} to have coefficients in
**Z**_{p}.
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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