Cohomology and Galois theory of 2-groups of maximal class

Abstract

The (finite) nonabelian \(2\)-groups \(X\) of maximal class are (generalized) quaternion, dihedral or semidihedral. If the order \(|X|=2^{n+1}\), then \(X\) is a Schur cover of the dihedral group \(G=D_{2^n}\) of order \(2^n\)(where \(D_4=V\) is the four group when \(n=2\)), but there are four distinct cohomology classes in \(H^2(G, Z_2)\) resulting from Schur covers. We show that the quaternion group \(Q_{2^{n+1}}\) gives rise to a unique cohomology class in \(H^2(G, Z_2)\), and that the same holds for \(D_{2^{n+1}}\) if and only if \(n\ge 3\). The results will be applied to Galois field extensions admitting such groups.