Cohomology and Galois theory of 2-groups of maximal class

摘要

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 ≥ 3. The results will be applied to Galois field extensions admitting such groups.