On the cohomology of finite soluble groups. (Q494605)

In this paper, a theorem of \textit{C. Parker} and \textit{P. Rowley} [Bull. Lond. Math. Soc. 42, No. 3, 417-419 (2010; Zbl 1193.20019)] is revisited and two vanishing theorems (Theorems 1 and 3) for group cohomology at dimensions 1 and 2 are proved. The author remarks that the vanishing theorem at dimension 1 is what is behind the Parker-Rowley theorem. Finally, some applications of these theorems are given. If \(\Omega\) is a set of unary operations on a group \(G\), then \(\mathrm{comp}_\Omega(G)\) denotes the set of isomorphism classes of \(\Omega\)-composition factors of \(G\). The Parker-Rowley theorem states the following: Suppose that \(N\) is a nilpotent normal subgroup of a finite solvable group \(G\) such that \(\mathrm{comp}_G(G/N)\cap\mathrm{comp}_G(N)\) is empty. If \(H\) and \(K\) are subgroups of \(G\) such that \(HN=KN=G\) and \(H\cap N=K\cap N\), then \(H\) and \(K\) are conjugate in \(G\). Motivated by this theorem, the following cohomological vanishing theorem is proved: Theorem 1. Let \(Q\) be a finite group and \(A\) be a finite \(Q\)-module. Assume that \(Q\) is \(p\)-solvable for all the primes \(p\) dividing the order of \(A\), and that \(\mathrm{comp}_Q(Q)\cap\mathrm{comp}_Q(A)\) is empty. Then \(H^1(Q,A)=0\). Using this theorem, a generalization of the Parker-Rowley theorem is also given (Theorem 2). Then in Section 3, a similar vanishing result for group cohomology at dimension 2 is proved (Theorem 3). The assumptions of Theorem 3 are too technical to state here. The paper ends with some applications of these theorems to the splitting of group extensions and to the structure of automorphism groups of finite solvable groups with nilpotent length at most \(3\).