On the descending central sequence of absolute Galois groups

Abstract

Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th 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 $G_F$ is the intersection of all open normal subgroups $N$ such that $G_F/N$ is 1, $\mathbb{Z}/p^2$, or the modular group $M_{p^3}$ of order $p^3$.

We implemented the referee's comments. The paper will appear in The American Journal of Mathematics

References in corpus (1)

• The mod-p cohomology rings of some p-groups

Cited by in corpus (2)

• Almost-Commuting-Liftable Subgroups of Galois Groups
• Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture