アルティン・ハッセの指数関数
(1) If G is the p-adic integers, it has exactly one open subgroup of each p-power index and a continuous homomorphism from G to Sn is essentially the same thing as choosing an element of p-power order in Sn, so we have recovered the above combinatorial interpretation of the Taylor coefficients in the Artin–Hasse exponential series. (2) If G is a finite group then the sum in the exponential is a finite sum running over all subgroups of G, and continuous homomorphisms from G to Sn are simply homomorphisms from G to Sn. The result in this case is due to Wohlfahrt (1977). The special case when G is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and takes the form
exp  ( ∑ d | m x d / d ) = ∑ n ≥ 0 a m , n n !