A characterization of the cyclic groups by subgroup indexes

LetG be a group. ThenG is cyclic if there exists some g ∈ G such thatG = 〈g〉 := {g : m ∈ Z}. For every positive integer n, Z/〈n〉 (the additive group of integers modulo n) is the unique cyclic group on n elements, and Z is the unique infinite cyclic group (up to isomorphism). The cyclic groups play a nontrivial role in abelian group theory. For instance, The Fundamental Theorem of Finitely Generated Abelian Groups states that every finitely generated abelian group is a finite direct sum of cyclic groups (see Hungerford [7], Theorem 2.1). Further, every abelian group G for which there is a finite bound on the orders of the elements of G is a (possibly infinite) direct sum of cyclic groups (cf. Fuchs [6], Theorem 11.2). Given the fundamental role the cyclic groups play in group theory, it is hardly a surprise that many characterizations of these groups have appeared in the literature over the years; see the bibliography for a sample of such papers. The purpose of this note is to present a new characterization via subgroup indexes. Recall that if G is a group and H < G (that is, H is a subgroup of G), then the index (G : H) of H in G is simply the cardinality of the set of right cosets of H in G; more compactly, (G : H) = |{Hg : g ∈ G}| (equivalently, (G : H) is the cardinality of the set of left cosets of H in G). It is not hard to show that distinct subgroups of a finite cyclic group have distinct cardinalities (we will shortly present a proof of this assertion). It then follows immediately that distinct subgroups of a finite cyclic group G have distinct indexes in G. The same property is enjoyed by the infinite cyclic group Z of integers. To wit, every subgroup of Z is of the form 〈m〉 for some integer m ≥ 0. Note that if m and n are distinct positive integers, then m = (Z : 〈m〉) 6= n = (Z : 〈n〉). Further, (Z : {0}) = א0. Hence distinct subgroups of Z have distinct indexes in Z. In this paper, we show that the previous property enjoyed by the cyclic groups completely distinguishes them within the class of all groups. That is, we prove that an arbitrary group G is cyclic if and only if distinct subgroups of G have distinct indexes in G.