Students Ask the Darnedest Things: A Result in Elementary Group Theory

Abstract

Introduction In most elementary group theory courses, students are introduced to the Fundamental Theorem of Finite Abelian Groups (FT), which states that every finite abelian group is isomorphic to exactly one direct product of cyclic groups of prime power order. (These cyclic groups are called invariant factors.) As every cyclic group of order k is isomorphic to Zk, the additive group of integers mod k, FT asserts that every abelian group of order 12 is isomorphic either to Z4 X Z3 or to Z2 X Z2 X Z3. (I'll use the common abbreviation Zk to denote Z/kZ, with apologies to my fellow number theorists, who usually use this notation to denote the k-adic integers.) A standard exercise is to determine whether a given abelian group G of order 12 is isomorphic to Z4 X Z3 or Z2 X Z2 x Z3 by computing the orders of the elements of G. For instance, if G contains an element of order 12 (or 4, for that matter), it must be isomorphic to the cyclic group Z4 X Z3, and if G has more than 1 element of order 2, then it is isomorphic to Z2 X Z2 X Z3. Notice, however that every abelian group of order 12 contains exactly two elements of order 3. An important concept reinforced through such an exercise is that isomorphisms preserve order. Specifically, if f: G -> G' is a group isomorphism, then g and f(g) have the same order in their respective groups. Thus if two finite groups G and G' are isomorphic, they must have identical order structure (the same number of elements of each order). This article examines the converse question, posed by a student in a group theory course: