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A split embedding for groups of class three. (English) Zbl 1009.20007
Let $$G$$ be a finite group and $$G_n$$ be the $$n$$-th lower central subgroup of $$G$$. Denote by $$A_G$$ the augmentation ideal of the integral group ring $$\mathbb{Z} G$$. It is known [G. Losey, Can. J. Math. 26, 962-977 (1974; Zbl 0255.20026)] that if $$G$$ is a finite group of odd order, then the canonical homomorphism $$G_3/G_4\to A^3_G/A^4_G$$ is a split monomorphism. Using the cocycle machinery of I. B. S. Passi [J. Algebra 9, 152-182 (1968; Zbl 0159.31503)], in this paper the author gives a new proof of this result when $$G$$ is a finite $$p$$-group of class 3 with $$p>2$$.
##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20D15 Finite nilpotent groups, $$p$$-groups 20F14 Derived series, central series, and generalizations for groups 16S34 Group rings