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(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient conditionfor 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 valuationof the coefficients of the exponential of $S(z)$, assuming weaker conditions onthe coefficients of $S(z)$ than in Dwork's lemma. As applications, we provideseveral results concerning lower bounds on the $p$-adic valuation of the numberof permutation representations of finitely generated groups. In particular, wegive fairly tight lower bounds in the case of an arbitrary finite Abelian$p$-group, thus generalising numerous results in special cases that hadappeared earlier in the literature. Further applications include sufficientconditions for ultimate periodicity of subgroup numbers modulo $p$ for freeproducts of finite Abelian $p$-groups, results on $p$-divisibility ofpermutation numbers with restrictions on their cycle structure, and a curious"supercongruence" for a certain binomial sum.

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