ON THE DESCENDING CENTRAL SEQUENCE OF ABSOLUTE GALOIS GROUPS

Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group G(F) of F. Namely, the third subgroup G(F)((3)) in the descending p-central sequence of G(F)((3)) is the intersection of all open normal subgroups N such that G(F)/N is 1, Z/p(2), or the extra-special group M-p(3) of order p(3) and exponent p(2).