The Cantor subset of the unit interval [0, 1) is "large" in cardinality and also "large" algebraically, that is, the smallest subgroup of [0, 1) generated by the Cantor set (using addition mod 1 as the group operation) is the whole of [0, 1). In this paper, we show how to construct Cantor-like sets which are "large" in cardinality but "small" algebraically. In fact for the set we construct, the subgroup of [0, 1) that it generates is, like the Cantor set itself, nowhere dense in [0, 1).