Tensor product factorizations with a commutative factor of modular group algebras over prime fields correspond exactly to direct product decompositions of the group.
Lazard, Sur les groupes nilpotents et les anneaux de Lie , Annales scientifiques de l'\'Ecole Normale Sup\'erieure 3e s \'e rie, 71 (1954), no
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On commutative tensor factors of group algebras
Tensor product factorizations with a commutative factor of modular group algebras over prime fields correspond exactly to direct product decompositions of the group.