The structure of finite \(p\)-groups: Effective proof of the coclass conjectures
In 1980 C. R. Leedham-Green and M. F. Newman proposed a program for classifying finite \(p\)-groups using the coclass as the primary invariant. The coclass of a group of order \(p^ n\) and nilpotency class \(c\) is defined to be \(n-c\); a pro-\(p\) group of coclass \(r\) is an inverse limit of finite \(p\)-groups of coclass \(r\). Thus, groups of coclass 1 are exactly the groups of maximal class studied by N. Blackburn and many others. Let us now state the coclass conjectures, ordered by increasing strength (that is, each conjecture implies the previous ones). Conjecture E. Given \(p\) and \(r\), there are only finitely many isomorphism types of infinite soluble pro-\(p\) groups of coclass \(r\). Conjecture D. Given \(p\) and \(r\), there are only finitely many isomorphism types of infinite pro-\(p\) groups of coclass \(r\). Conjecture C. Pro-\(p\) groups of finite coclass are soluble. Conjecture B. For some function \(g\), every \(p\)-group of coclass \(r\) has derived length at most \(g(p,r)\). Conjecture A. For some function \(f\), every \(p\)-group of coclass \(r\) has a normal subgroup \(N\) of class at most 2 (1, if \(p=2\)) and index at most \(f(p,r)\). Some of these conjectures may be reformulated as combinatorial statements on the structure of certain graphs. Thus, let \(T_{p,r}\) be a (directed) tree whose vertices are all \(p\)-groups of coclass \(r\), where \((H,G)\) is an edge if there is an epimorphism from \(H\) to \(G\) whose kernel has order \(p\). Then Conjecture D is tantamount to saying that there are only finitely many infinite chains in \(T_{p,r}\). Over the past 15 years the coclass conjectures attracted the attention of various mathematicians, such as Leedham-Green, S. Donkin, S. McKay, W. Plesken, A. Mann, E. I. Zelmanov, M. F. Newman, E. A. O'Brien, and others. The combination of Donkin's and Leedham-Green's important contributions settles Conjecture A in the affirmative for primes \(p>3\). This approach, which is based on algebraic groups and inverse limit arguments, yields no bounds on the function \(f(p,r)\) appearing in Conjecture A. Recently, using a different (and more elementary) approach, Zelmanov and the author have obtained a short proof of Conjecture C for all primes \(p\). In the present paper Conjecture A is proved for all primes \(p\), with explicit bounds on \(f(p,r)\). More specifically we prove the following Theorem: Let \(G\) be a finite \(p\)-group of coclass \(r\), and let \(\{\gamma_ n\}\) be its lower central series. 1. Suppose \(p>2\). Then \(\gamma_{2(p^ r- p^{r-1} -1)}\) has class at most 2. Suppose \(p=2\) and \(|G|\geq 2^{2^{2r+5}}\). Then \(\gamma_{7.2^ r-2}\) is abelian. For odd primes this confirms Conjecture A with \(f(p,r)\leq p^{2(p^ r- p^{r-1} -1)+r -1}< p^{2p^ r}\), a bound which is quite realistic. The bound obtained for \(p=2\) is probably less accurate. In the first stage of the proof the theory of powerful \(p\)-groups and uniserial actions is used in order to reduce to a Lie-theoretic problem. That problem, which involves nonsingular derivations of modular Lie algebras, is then solved using classical theorems of Engel and Jacobson on Lie algebras satisfying the Engel condition. The case \(p=2\) is more subtle and requires some extra stages. The paper contains several additional results on the structure of groups of given coclass.
scientific article; zbMATH DE number 549848
scientific article