The solution of the \(k(GV)\)-problem.
Let \(H\) be a finite group and let \(p\) be a prime divisor of \(|H|\). Let \(B\) be a \(p\)-block of irreducible complex characters of \(H\) and let \(D\) be the defect group of \(B\). Let \(k(B)\) denote the number of irreducible characters of \(H\) in \(B\). Brauer's \(k(B)\)-conjecture is that \(k(B)\leq|D|\). In 1959, Brauer and Feit proved that \(k(B)\leq\tfrac 14|D|^2+1\) and this remains the best general bound obtained for \(k(B)\). While the \(k(B)\)-conjecture has remained fairly inaccessible, its verification for the family of \(p\)-solvable groups has seemed a reasonable goal, as one may hope to exploit the additional structure of such groups to force out a successful conclusion. The following is a typical problem faced when investigating such \(p\)-solvable groups. Let \(G\) be an arbitrary \(p'\)-group and let \(V\) be an elementary Abelian \(p\)-group on which \(G\) acts faithfully. Let \(GV\) denote the semi-direct product, where \(G\) acts naturally on the normal subgroup \(V\), and let \(k(GV)\) denote the number of conjugacy classes in \(GV\). The group \(GV\) is certainly \(p\)-solvable and elementary block theory shows that \(GV\) has only one \(p\)-block, namely the principal block \(B_1\), with defect group \(V\). Thus \(k(B_1)=k(GV)\) and, if correct, Brauer's conjecture implies that \(k(GV)\leq|V|\). Now in 1962, Nagao showed that, conversely, if \(k(GV)\leq|V|\) for all \(p'\)-groups \(G\) and all \(V\) on which \(G\) acts faithfully, then the \(k(B)\)-conjecture is true for all \(p\)-solvable groups. The \(k(GV)\)-problem is to prove the inequality \(k(GV)\leq|V|\). This paper is the conclusion of the proof that we do have \(k(GV)\leq|V|\), and thus the \(k(B)\)-conjecture is true for all \(p\)-solvable groups. It has taken just over 40 years since the appearance of Nagao's paper for the combined efforts of several mathematicians to be crowned by the final success of the four authors of this paper. It was not until the 1980's that any substantial progress was made on the \(k(GV)\)-problem, when Knörr showed that \(k(GV)\leq|V|\) if \(G\) is supersolvable or if \(|G|\) is odd. He also showed the importance of trying to find a (near) square root for the permutation character of \(G\) acting on the elements of \(V\). A (near) square root exists when \(V\) is self-dual as a \(G\)-module. This idea was exploited with great effect in a paper of 1996 by Robinson and Thompson, who showed that \(k(GV)\leq|V|\) if there is some \(v\in V\) such that \(V\) contains a faithful self-dual \(C_G(v)\)-submodule. Such vectors \(v\) are called real vectors in this situation. The advantage of using real vectors (when they exist) is that they are amenable to the usual Clifford analysis of modules by means of normal subgroups. Exploiting a variety of techniques, including the classification of finite simple groups, Robinson and Thompson were able to prove that real vectors exist provided \(p>5^{30}\) and hence the \(k(B)\)-conjecture is true for \(p\)-solvable groups when \(p\) satisfies this inequality. Eventually, a difficult analysis, conducted by several independent researchers, showed that real vectors exist unless \(p\) is one of 3, 5, 7, 11, 13, 19 or 31. These primes are exceptional as we may use them to construct examples of suitable \(G\) and \(V\) for which no real vectors exist. Since the year 2000, the authors of this paper, working independently, or as the two pairs Gluck and Magaard, and Riese and Schmid, have managed to eliminate all these exceptional primes, apart from 5. In this paper, they have combined their resources to dispose of the prime 5 and thereby solve the \(k(GV)\)-problem. Their proof requires much ingenuity, as the arguments involve the analysis of several tight configurations of group actions, and the real vector arguments cannot be applied. An interesting subsidiary result which the authors need to prove is that, if \(S\) is a primitive \(5'\)-group of degree \(n>20\), then the number of conjugacy classes in \(S\) is at most \(2^{n/5}\). The proof of this result utilizes many of the weapons in the arsenal of modern finite group theory. In conclusion, the authors have made a major contribution to the study of block theory and have demonstrated just how delicate and challenging some of the original conjectures in the subject have turned out to be.
scientific article; zbMATH DE number 2104459
scientific article