The positive solution by N. Nikolov and D. Segal of Serre’s problem as to whether every finite index subgroup of a finitely generated profinite group is open created a new direction in the study of profinite groups. Namely the study of the abstract subgroup structure of a profinite group. The absence of such a study before can be explained by the fact that from the point of view of Galois theory it does not make much sense, since non-closed subgroups of a Galois group are not Galois groups of intermediate extensions; profinite group theory was born from the Galois theory, namely as the study of Galois groups of infinite field extensions, a fact that determined the main lines of research in the area.
The present paper is an outstanding progress in the study of abstract subgroup structure of profinite groups.
In this paper the authors obtain important new results in the study of the abstract normal subgroup structure of profinite groups and apply their new methods to strengthen and to give a new, shorter and more streamlined proof of their results on the solution of Serre’s problem and on verbal subgroups of profinite groups [Ann. Math. (2) 165, No. 1, 171-238, 239-273 (2007; Zbl 1126.20018); Groups Geom. Dyn. 5, No. 2, 501-507 (2011; Zbl 1243.20036)].
We start listing the main results for profinite groups of the paper.
Let \(G\) be a finitely generated profinite group. Put \(G_0=\bigcap_T\), where \(T\) ranges over all open normal subgroups such that \(G/T\) is finite almost simple (a finite group \(H\) is almost simple if \(S\triangleleft H\leq\operatorname{Aut}(S)\) for some non-Abelian simple group \(S\)). Denote by \(d(G)\) the minimal number of generators of \(G\).
Theorem 1.5. Let \(G\) be a finitely generated profinite group and \(K\leq G_0\) a closed normal subgroup of \(G\). Suppose that \(G=K\overline{\langle y_1,\dots,y_r\rangle}=\overline{G'\langle y_1\dots,y_r\rangle}\). Then there exist elements \(x_{ij}\in K\) such that \[ G=\overline{\langle y_i^{x_{ij}}\mid i=1,\dots,r,\;j=1,\dots,f_0\rangle} \] where \(f_0=f_0(r,d(G))\) and \(G'\) means the abstract commutator.
Note that the quotient group \(G/G_0\) is an extension of a finite semisimple group by a soluble profinite group of derived length at most 3.
To formulate the next results we use the following notation from the paper. For a subset \(X\) of \(G\), we write \(X^{*f}= \{x_1x_2\cdots x_f\mid x_1,x_2,\dots,x_f\in X\}\). The subset \(X\) is symmetric if \(x\in X\) implies \(x^{-1}\in X\).
Theorem 1.6. Let \(G\) be a profinite group and \(\{y_1,\dots,y_r\}\) be a symmetric set of topological generators of \(G\). If \(H\) is a closed normal subgroup of \(G\) then \[ [H,G]=(\prod_{i=1}^r[H,y_i])^{*f_1} \] where \(f_1=f_1(r, d(g))\).
Theorem 1.7. Let \(G\) be a finitely generated profinite group, \(H\leq G_0\) a closed normal subgroup of \(G\) and \(\{y_1,\dots,y_r\}\) a symmetric subset of \(G\). If \(H\overline{\langle y_1,\dots,y_r\rangle}=G\) then \[ [H,G]=(\prod_{i=1}^r[H,y_i])^{*f_2} \] where \(f_2=f_2(r,d(G))\).
The heart of the proofs of the theorems lies in the case that \(G\) is finite. Theorem 1.7 in this case fundamentally improves the key Theorem C from [loc. cit., Zbl 1126.20018] eliminating its dependence on one more parameter (certain measure of the complexity of \(G\)). Moreover, the authors show that in this case \(f_0=O(rd(G)^2)\), \(f_1=O(r^2d(G))=O(r^3)\), \(f_2=O(r^6d(G)^6)\).
As a consequence it is shown in the paper that if \(G\) is profinite finitely generated, then for any abstract proper normal subgroup \(N\) of \(G\) either \(NG'\) or \(NG_0\) is proper in \(G\). This is the key for understanding abstract normal subgroups of \(G\) since \(G/G'\) and \(G/G_0\) have relatively transparent structure.
The last section of the paper is dedicated to compact groups to which the authors extend many of the results above. Namely, suppose \(G\) is a compact topological group, \(G^0\) its connected component of \(1\) and \(G/G^0\) a finitely generated profinite group.
Theorem 1.10. Let \(G\) be a semisimple compact topological group that is either finitely generated profinite or connected. If \(Q\) is an infinite (abstract) quotient then it is not countable.
A group is called ‘Fab’ if every virtually Abelian quotient is finite.
Theorem 5.23. Let \(G\) be a compact topological group such that \(G/G^0\) is topologically finitely generated. Then every countable Fab quotient of \(G\) is finite.
As a very nice corollary it is deduced that \(G\) has countably infinite quotient if and only if \(G\) is not Fab.
Theorem 1.13. Let \(G\) be a compact group and \(N\) an abstract normal subgroup of \(G\) such that \(G/N\) is (abstractly) finitely generated. Then \(G/N\) is finite.
The next result treats the existence of a virtually dense normal subgroup of infinite index.
Theorem 1.14. Let \(G\) be a compact group such that \(G/G^0\) is (topologically) finitely generated. Then \(G\) has a virtually dense normal subgroup of infinite index if and only if some open normal subgroup of \(G\) has an infinite Abelian quotient or a strictly infinite semisimple quotient.
It is deduced as an easy but astonishing consequence of this that a finitely generated just infinite not virtually Abelian profinite group has only closed normal subgroups.