In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field Fq (x) whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥ 3, we show that for every ∈ > 0, there are » qL(1/2+ 3/ 2(g+1) -∈) polynomials f ∈ Fq [x] with deg f = L, for which the class group of the quadratic extension Fq (x, √ f) has an element of order g. This sharpens the previous lower bound qL(12 +1 g) of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields. © Indian Academy of Sciences.