Applications of the Artin-Hasse Exponential Series And its Generalizations to Finite Algebra Group

If F is a finite field of characteristic p and order q and J is a finite-dimensional nilpotent associative F-algebra, then we call the finite p-group G = 1+J an F-algebra group. A subgroup H of G is called an algebra subgroup if H = 1+A for some subalgebra A of J. A subgroup K of G is said to be strong if the order of the intersection of K with H is a power of q for all algebra subgroups H of G. If Jp = 0, then the ordinary exponential series can be used to show that normalizers of algebra subgroups are strong. If Jp is not equal to 0, then the exponential series does not make sense, but a generalization, the Artin-Hasse exponential series, is defined. We use the Artin-Hasse series to determine when normalizers of algebra subgroups are strong and when counter-examples exist. In addition, we give a description of strong subgroups in terms of stringent power series, that is, power series whose linear coefficient and constant term are both 1.