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# Slobodan Tanusevski: "Frattini-resistant pro-p groups"

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Description

Abstract: A profinite group is a compact and totally disconnected topological group.

The Galois group of an infinite Galois extension of fields, equipped with the Krull topology, is a profinite group.

Conversely, every profinite group can be realized as Galois groups.

Let p be a prime number. A pro-p group is a profinite group in which every open subgroup has index some power of p. The additive group of the ring of p-adic integers is an example of a pro-p group.

All maximal subgroups of a pro-p group G are normal and have index p; their intersection is called the Frattini-subgroup of G.

Let G be a pro-p group. Consider the function that maps each subgroup H of G to the Frattini-subgroup of H.

If this function is injective, then we say that G is a Frattini-injective pro-p group.

Let K be a field containing a primitive pth root of unity, and let K(p) denote the maximal p-extension of K, that is, the compositum of all finite

Galois p-extensions of K inside a fixed separable closure of K. The Galois group Gal(K(p)/K) is called the maximal pro-p Galois group of K.

It is an example of a Frattini-injective pro-p group. In fact, the notion of Frattini-injectivity seems to capture an essential property of maximal pro-p Galois groups.

Indeed, a lot of what is known about maximal pro-p Galois groups can be deduced (without recourse to "heavy machinery") from Frattini-injectivity.

This is a joint work with Ilir Snopce.