A counterexample to the isomorphism problem for integral group rings.

*(English)*Zbl 0990.20002This is a landmark paper! The author finds in this masterpiece the complete solution to the 60 year old problem posed for the first time in the thesis of Graham Higman. Let \(G\) and \(H\) be finite groups. Can one conclude that if the integral group rings \(\mathbb{Z} G\) and \(\mathbb{Z} H\) are isomorphic as rings, then \(G\) and \(H\) are isomorphic as groups? In the present paper Martin Hertweck produces explicitly two non-isomorphic finite groups \(G\) and \(H\) with isomorphic integral group rings \(\mathbb{Z} G\simeq\mathbb{Z} H\). So, the above question is answered to the negative. This result brings a long history to an end. The main steps are sketched below.

Higman asked this question in his 1940 thesis and answered it to the positive for Abelian groups. Glauberman proved in 1965 that by means of an isomorphism of the integral group rings of \(G\) and \(H\) the normal subgroup structure of \(G\) and of \(H\) can be identified. Using this, Whitcomb could answer the question to the positive for finite metabelian groups in his 1968 thesis. The subject has undergone a tremendous progress when K. W. Roggenkamp and L. L. Scott proved a series of results with entirely new methods. Amongst many other things, they answered in 1986 the question to the positive for finite \(p\)-groups and for finite nilpotent groups [Ann. Math. (2) 126, 593-647 (1987; Zbl 0633.20003)]. Actually, they proved a much stronger statement on the conjugacy structure of the units in the group ring. This proved to be the key idea in the subject. In 1993 K. W. Roggenkamp and the reviewer constructed [J. Pure Appl. Algebra 103, No. 1, 91-99 (1995; Zbl 0835.16020)] a group \(G\) having an automorphism which is not inner in \(G\) but which becomes inner in the group ring \(RG\) for any semilocalization \(R\) of \(\mathbb{Z}\). One year later and independently, Marcin Mazur studied [Expo. Math. 13, No. 5, 433-445 (1995; Zbl 0841.20011)], under some hypotheses on \(R\), this isomorphism problem for groups being a direct product of a finite group \(G\) and an infinite cyclic group \(C_\infty\). He proved that \(R(G\times C_\infty)\simeq RW\) if and only if \(W\) is a semidirect product of a finite group \(H\leq W\) with an infinite cyclic group acting on \(H\) by an automorphism which becomes inner in \(RH\), and where \(RG\simeq RH\).

These last two results [Zbl 0835.16020] and [Zbl 0841.20011] are the starting point of Martin Hertweck’s paper. Martin Hertweck ingeniously analyzed the paper [Zbl 0835.16020] and realized that the non-constructive part of the machinery used there is not really necessary. He observed that one can generalize Mazur’s observation to considering two semidirect products of a fixed finite group \(H\) with finite cyclic groups, and then the group rings are isomorphic if the actions differ by an automorphism which is inner in \(RH\). Then, using a mixture of arithmetic constraints and the algebraic considerations he is able to construct the group \(H\) on which a finite cyclic group acts in two different ways. In order to prove that the two groups are not isomorphic, Martin Hertweck uses some Čech style cohomology technique.

This wonderfully well written paper not only gives the complicated construction, but also it explains it so well, that one is able to read it fluently. The very involved construction of the groups is presented in a concise and precise way. The key steps of the proofs are clearly marked and organized. The paper reflects the author’s great insight into the subject.

Higman asked this question in his 1940 thesis and answered it to the positive for Abelian groups. Glauberman proved in 1965 that by means of an isomorphism of the integral group rings of \(G\) and \(H\) the normal subgroup structure of \(G\) and of \(H\) can be identified. Using this, Whitcomb could answer the question to the positive for finite metabelian groups in his 1968 thesis. The subject has undergone a tremendous progress when K. W. Roggenkamp and L. L. Scott proved a series of results with entirely new methods. Amongst many other things, they answered in 1986 the question to the positive for finite \(p\)-groups and for finite nilpotent groups [Ann. Math. (2) 126, 593-647 (1987; Zbl 0633.20003)]. Actually, they proved a much stronger statement on the conjugacy structure of the units in the group ring. This proved to be the key idea in the subject. In 1993 K. W. Roggenkamp and the reviewer constructed [J. Pure Appl. Algebra 103, No. 1, 91-99 (1995; Zbl 0835.16020)] a group \(G\) having an automorphism which is not inner in \(G\) but which becomes inner in the group ring \(RG\) for any semilocalization \(R\) of \(\mathbb{Z}\). One year later and independently, Marcin Mazur studied [Expo. Math. 13, No. 5, 433-445 (1995; Zbl 0841.20011)], under some hypotheses on \(R\), this isomorphism problem for groups being a direct product of a finite group \(G\) and an infinite cyclic group \(C_\infty\). He proved that \(R(G\times C_\infty)\simeq RW\) if and only if \(W\) is a semidirect product of a finite group \(H\leq W\) with an infinite cyclic group acting on \(H\) by an automorphism which becomes inner in \(RH\), and where \(RG\simeq RH\).

These last two results [Zbl 0835.16020] and [Zbl 0841.20011] are the starting point of Martin Hertweck’s paper. Martin Hertweck ingeniously analyzed the paper [Zbl 0835.16020] and realized that the non-constructive part of the machinery used there is not really necessary. He observed that one can generalize Mazur’s observation to considering two semidirect products of a fixed finite group \(H\) with finite cyclic groups, and then the group rings are isomorphic if the actions differ by an automorphism which is inner in \(RH\). Then, using a mixture of arithmetic constraints and the algebraic considerations he is able to construct the group \(H\) on which a finite cyclic group acts in two different ways. In order to prove that the two groups are not isomorphic, Martin Hertweck uses some Čech style cohomology technique.

This wonderfully well written paper not only gives the complicated construction, but also it explains it so well, that one is able to read it fluently. The very involved construction of the groups is presented in a concise and precise way. The key steps of the proofs are clearly marked and organized. The paper reflects the author’s great insight into the subject.

Reviewer: Alexander Zimmermann (Amiens)

##### MSC:

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

16S34 | Group rings |

16U60 | Units, groups of units (associative rings and algebras) |

20C10 | Integral representations of finite groups |