This note identifies an error in Exercise 8 on page 306 of the second edition of “Introduction to Finite Mathematics,” by Kemeny, Snell and Thompson. That exercise incorrectly stated that if a and b are distinct elements of a finite group G, then the subgroup of G that is generated by a and b is cyclic if and only if at least one of the elements a and b is a power of the other element. We give the smallest counterexample G to that incorrect exercise and we then give several characterizations of the finite groups G for which the assertion in the above-mentioned exercise is valid. This note could be used as enrichment material for classes familiar with the rudiments of group theory.

finite group, cyclic group, subgroup, Lagrange’s Theorem, abelian group, prime number, Chinese Remainder Theorem, symmetric group.