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Unformatted text preview: J. Group Theory 6 (2003), 223-228 Journal of Group Theory cg de Gruyter 2003 The unit group of 1 + A(G)A(A) is torsion-free Zbigniew Marciniak and Sudarshan K. Sehgal* (Communicated by I. B. S. Passi) Abstract. Let A be an abelian normal subgroup of a finite group G. Then all units of the inte- gral group ring 7lG which are of the form 1+ J with 0 ""J E d( G)d(A), where d denotes the augmentation ideal, are of infinite order. 1 Introduction Let TLG be the integral group ring of a group G. Graham Higman asked in 1940, in , the following question. The IsomorphismProblem. Suppose that the rings TLG and TLH are isomorphic. Does it follow that the groups G andH are isomorphic? Today we know that for some classes of groups the answer is positive; see [9, p. 207] for an account. On the other hand, Hertweck  has recently constructed two (large) non-isomorphic finite groups with isomorphic integralgroup rings. However, the situation is still far from being completely understood. Higman himself positively solved this problem for finite abelian groups A, by identifying + A with the set of torsion elements in the group O//(7LA) of units of 7LA. He proved that O//(7LA) = :t:A x all(1 + .1(A)2), where .1(A) is the kernel of the augmentation homomorphism e: TLA-4 7L defined by e(~ ngg) = ~ ng. Higman proved that the subring 1 + .1(A)2 has no torsion units, hence the subgroup A < O//(7LA) has a torsion-free normal complement. Higman's description of units in 7lA was generalized in  for groups G with a normal abelian subgroup A such that G / A is abelian of exponent 2, 3, 4 or 6. This time O//(7LG) = +G. all(1 + .1(G).1(A)), *This research was supported by NSERC Canada and Polish KBN Grant No. 2PO3AO0218.-- ~~- 224 Zbigniew Marciniak and Sudarshan K. Sehgal where ~(G)~(A) is the two-sided ideal in 7lG generated by all elements of the form (g- I)(a - I) with g E G, a E A. Again, the subring I + ~(G)~(A) has no torsion units. It is easy to show that in general, if a finite group G has a torsion-free normal complement in the unit group rJ/i(71G) and 7lG ~ 7lH then G ~ H. Hence it was asked by Dennis whether the embedding G --* rJ/i(7LG) always splits and the nor- mal complement arising as the kernel of the splitting homomorphism is torsion- free. Roggenkamp and Scott  constructed counter-examples to this even for meta- belian groups. On the other hand, Cliff, Sehgal and Weiss [I] gave an affirmative answer for metabelian groups A <J G--* G / A with I G / A I odd. Again, normal com- plements arise from ideals related to ~(G)~(A). Moreover, they proved that the group rJ/i(1+ il(G)il(A)) is torsion-free for...
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