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On the commutativity of the radical of a group algebra
Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 1-8

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Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative. 
Wallace, D. A. R. On the commutativity of the radical of a group algebra. Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 1-8. doi: 10.1017/S2040618500035061
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