Geodesic
Group algebras with radicals of square zero
Glasgow mathematical journal, Tome 5 (1962) no. 4, pp. 158-159

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Over a field of 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. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself. 
Wallace, D. A. R. Group algebras with radicals of square zero. Glasgow mathematical journal, Tome 5 (1962) no. 4, pp. 158-159. doi: 10.1017/S2040618500034523
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