Geodesic
Group structure and properties of block ideals of the group algebra
Glasgow mathematical journal, Tome 16 (1975) no. 1, pp. 22-28

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In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:(i) J(B1) ≤ Z(B1)(ii) J(B1)is commutative,(iii) G is p-nilpotent with abelian Sylowp-subgroups. 
Hamernik, Wolfgang. Group structure and properties of block ideals of the group algebra. Glasgow mathematical journal, Tome 16 (1975) no. 1, pp. 22-28. doi: 10.1017/S0017089500002457
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