On the restriction map in the mod-2 cohomology of groups

Abstract

Marx (1990) has proved that the kernel of the restriction map from H ∗(G,Z 2) to all proper subgroups is of nilpotence degree 2, provided that G is a 2-group generated by two elements, or the Frattini subgroup of G is central and is not elementary abelian. We prove that this fact also holds for 3-generator 2-groups G satisfying one of the following conditions: (i) ¦G¦≤ 2 6 ; (ii) the Frattini subgroup of G is central; (iii) G is an extension of either an abelian group, or D 8 × C 2, or Q 8 × C 2.