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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 \(p>2\).
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