Finite axiomatizability for profinite groups

Maximal Subgroups in Finite and Profinite Groups

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann....

Almost Engel finite and profinite groups

Subgroups of Finite Index in Profinite Groups

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...

Profinite Groups

γ = c0 + c1p+ c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it...

Normal subgroups of profinite groups of finite cohomological dimension

We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N). Some other cases when G/N has virtually finite cohomological dimension are considere...