The isomorphism problem for group rings: A survey.

*(English)*Zbl 0565.20005
Orders and their applications, Proc. Conf., Oberwolfach/Ger. 1984, Lect. Notes Math. 1142, 256-288 (1985).

[For the entire collection see Zbl 0561.00005.]

The paper under review provides an extensive survey on the isomorphism problem for group rings. It is well written, very informative and touches almost every aspect of the subject. Especially impressive is the bibliographical material compiled by the author. The content of the paper is best described in the following quotation:

”After a section of preliminaries, the survey begins in Section 3 which outlines the conventional corpus of the subject with its emphasis on soluble groups: The characterization of the lattice of normal subgroups; the determination of metabelian groups. Section 4 concentrates on application of these results to non-soluble groups, showing, for example, that the groups PSL(n,q), SL(n,q) and GL(n,q) are all determined by their integral group rings. The fifth section describes a variety of different contexts in which the problem may be examined.... The last section of the text explores the problem for other rings of coefficients.”

The author supplied a list of errata from which we quote some major corrections: ”3.24 p. 262. As pointed out by Leonard Scott, the justification for this assertion is inadequate. The problem lies in the action of the Steenrod algebra on the cohomology ring. It is not clear that this is an invariant of \({\mathbb{Z}}G\) as ring. The cited presentation in Ben 84 uses the Hopf algebra structure of \({\mathbb{Z}}G\). 6.20 p. 272. The preprint in which this appears has been withdrawn. The result is in fact incorrect; \(C_ p\times C_ p\), p odd, provides a counterexample. The omission of this result necessitates the following alterations: 6.21. Delete. 6.22(ii), Change \(2n+2\) to \(2n+1\); 6.22(iii) Delete.”

The paper under review provides an extensive survey on the isomorphism problem for group rings. It is well written, very informative and touches almost every aspect of the subject. Especially impressive is the bibliographical material compiled by the author. The content of the paper is best described in the following quotation:

”After a section of preliminaries, the survey begins in Section 3 which outlines the conventional corpus of the subject with its emphasis on soluble groups: The characterization of the lattice of normal subgroups; the determination of metabelian groups. Section 4 concentrates on application of these results to non-soluble groups, showing, for example, that the groups PSL(n,q), SL(n,q) and GL(n,q) are all determined by their integral group rings. The fifth section describes a variety of different contexts in which the problem may be examined.... The last section of the text explores the problem for other rings of coefficients.”

The author supplied a list of errata from which we quote some major corrections: ”3.24 p. 262. As pointed out by Leonard Scott, the justification for this assertion is inadequate. The problem lies in the action of the Steenrod algebra on the cohomology ring. It is not clear that this is an invariant of \({\mathbb{Z}}G\) as ring. The cited presentation in Ben 84 uses the Hopf algebra structure of \({\mathbb{Z}}G\). 6.20 p. 272. The preprint in which this appears has been withdrawn. The result is in fact incorrect; \(C_ p\times C_ p\), p odd, provides a counterexample. The omission of this result necessitates the following alterations: 6.21. Delete. 6.22(ii), Change \(2n+2\) to \(2n+1\); 6.22(iii) Delete.”

Reviewer: G.Karpilovsky

##### MSC:

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

16S34 | Group rings |

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20D30 | Series and lattices of subgroups |