On the number of groups of a given order

AbstractLetting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ϕ(n) and G(n) ≤ (log n)c on a set of positive density. Letting Fk(x) denote the number of n ≤ x for which G(n) = k, it is shown that F2(x) = O(x(log4x)(log3x)2), where logrx denotes the r-fold iterated logarithm

On Kummer's Conjecture

AbstractKummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes

On Artin's conjecture

AbstractLet F be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of F. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on F, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if F is the family of fields obtained by adjoining to Q the q-division points of an elliptic curve E over Q, the Artin problem determines how often E(Fp) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields

On the density of various classes of groups

AbstractLet T be a subset of the set of all isomorphism classes of finite groups. We consider the number Fg(x) of positive integers n≤x such that all groups of order n lie in T. When T consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for Fg(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable groups. In the course of the arguments, we also obtain, for almost all n, a lower bound for the number of groups of a given order n

On the enumeration of finite groups

AbstractLet G(n) denote the number of finite groups of order n (up to isomorphism). We prove that for n squarefree, G(n) = Ω(n1−ε) for any ε > 0, and that for almost all squarefree integers n, log G(n) = (1 + o(1))(log log n) Σp¦n(log p)(p − 1). If we let Fk(x) be the number of n ≤ x such that G(n) = k, then we prove Fk(x) = (c(a) + o(1)) x(log log log x)a + 1 for k = 2a, and c(a) is an appropriate constant, as x → ∞. If k ≠ 2a, then we show that Fk(x) = O(x(log log x)1 − ε)