On the number of groups of a given order

Abstract

Letting 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 F k ( x) denote the number of n ≤ x for which G( n) = k, it is shown that F 2(x) = O(x( log 4x) ( log 3x) 2) , where log r x denotes the r-fold iterated logarithm.