The Baer-invariant of a Semidirect Product

الأصل · EN

In 1972 K.I.Tahara [7,2 Theorem 2.2.5], using cohomological method, showed that if a finite group G=T<N is the semidirect product of a normal subgroup N and a subgroup T, then M(T) is a direct factor of M(G), where M(G) is the Schur-multiplicator of G and in the finite case, is the second cohomology group of G. In 1977 W.Haebich [1 Theorem 1.7] gave another proof using a different method for an arbitrary group G. In this paper we generalize the above theorem. We will show that NcM(T) is a direct factor of NcM(G), where Nc [3 page 102] is the variety of nilpotent groups of class at most c≥ 1 and NcM(G) is the Baer-invariant of the group G with respect to the variety Nc [3 page 107].