Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Truncated versions of Dwork's lemma for exponentials of power series and
  $p$-divisibility of arithmetic functiens   (Dieudonn\'e 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.

1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys