The main result of the paper asserts that for any finite abelian group \(G\) and generating subset \(S\) such that \(0\not\in S\) and \( | S | \geq 5\), the number of elements representable as a sum of certain elements of \(S\) is \(\geq \min ( | G | -1, 2 | S |)\). The same holds for noncommutative groups if rearrangement of the elements is permitted, and the \(-1\) can be removed for cyclic groups. These results are best possible. Similar estimates were obtained previously under further assumptions by J. E. Olson [Acta Arith. 28, 147-156 (1975; Zbl 0318.10035)] and E. T. White [J. Comb. Theory, Ser. A 24, 118-121 (1978; Zbl 0369.20021)].