Profinite Groups

Free-by-Demushkin pro-p groups

“…We claim that the right vertical map is the zero one. Indeed by [10,Lemma 7.4.1] the rows of the following commutative diagram (of finite abelian groups of exponent p) are isomorphisms…”

Profinite Groups With Finite Virtual Length

“…We recall that, for a given prime p, a p-Sylow subgroup of a profinite group is a pro-p subgroup which is maximal in the family of pro-p subgroups. If G is a pronilpotent group then it has, for every prime p, a unique p-Sylow subgroup (which is then topologically fully invariant) and G is the cartesian product of its Sylow subgroups (see [9,Proposition 2.3.8]). E 2.10.…”

Alternating forms and the Brauer group of a geometric field

“…By the structure theory ( [RZ,Theorem 4.3.3,Theorem 4.3.9]), there is a topological isomorphism G I G i , where I is a set, G i is procyclic, and I G i has the product topology. In the torsion-free case, G i = Z m i , for some m i ⊂ P, and in the torsion case G i = Z/d i Z, for some d i ∈ N that divides the exponent.…”

“…In the torsion-free case, G i = Z m i , for some m i ⊂ P, and in the torsion case G i = Z/d i Z, for some d i ∈ N that divides the exponent. By [RZ,Lemma 4.3.7], a torsion proabelian group has bounded exponent. If G i Z m i , φ ∈ G i , and c ∈ Z m i , then φ c is the element whose image in each finite quotient The guiding example for the definition below arises when G is isomorphic to a finite homogeneous product of torsion-free procyclic groups, all of the same type, such as Z n .…”