Orders for Which There Exist Exactly Two Groups

Introduction. A long-standing problem in group theory is to determine the number of non-isomorphic groups of a given order. The inverse problem–determining the orders for which there are a given number of groups–has received considerably less attention. In this note, we will give a characterization of those positive integers n for which there exist exactly 2 distinct groups of order n (up to isomorphism). We call such numbers 2-group numbers. Our work extends ideas from [PS]. Following the authors, we say a number n = p1 1 · · · par r has nilpotent factorization if pi 6≡ 1 (mod pj) for all integers i, j, and k with 1 ≤ k ≤ ai. In other words, no prime power in the factorization is congruent to 1 modulo any other prime in the factorization. [PS] show in their paper that a number n has nilpotent factorization if and only if every group of order n is nilpotent. Such n they call, naturally enough, nilpotent numbers. Cyclic numbers are those n for which every group of order n is cyclic, and similarly for abelian numbers. The authors show that the cyclic numbers are precisely the square-free nilpotent numbers, and the abelian numbers are precisely the cube-free nilpotent numbers. We will see that every 2-group number is either a square-free number that fails nilpotent factorization for exactly one pair of primes, or an abelian number with exactly one square in its prime factorization.