The ring of symmetric functions is used to obtain an explicit set of generators for the centre of the integral group algebra of a symmetric group, different to those given by H.K. Farahat and G. Higman. This generating set is used to shows that the centre of the 2-modular group algebra is generated by certain sums of 2-classes. Proper subalgebras of the centre of the 2-modular group algebra are studied in the context of symmetric functions. These include the algebra that is the span of the 2-regular class sums, and the algebra that is generated by the involution class sums. Various related subalgebras of the modular ring of symmetric functions are shown to be polynomial algebras, and furnished with explicit sets of algebraically independent generators.