アルティン・ハッセの指数関数
The Artin–Hasse exponential is the generating function for the probability a uniformly randomly selected element of Sn (the symmetric group with n elements) has p-power order (the number of which is denoted by tp,n):
E p ( x ) = ∑ n ≥ 0 t p , n n ! x n . {\displaystyle E_{p}(x)=\sum _{n\geq 0}{\frac {t_{p,n}}{n!}}x^{n}.}
This gives a third proof that the coefficients of Ep(x) are p-integral, using the theorem of Frobenius that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d. Apply this theorem to the nth symmetric group with d equal to the highest power of p dividing n!