Small profinite groups

2001 ◽  
Vol 66 (2) ◽  
pp. 859-872 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Abelian Subgroup ◽  
Profinite Group ◽  
Profinite Groups

AbstractWe propose a model-theoretic framework for investigating profinite groups. Within this framework we define and investigate small profinite groups. We consider the question if any small profinite group has an open abelian subgroup.

Maximal abelian subgroups of free profinite groups

1985 ◽  
Vol 97 (1) ◽  
pp. 51-55 ◽  
Author(s):  
Dan Haran ◽  
Alexander Lubotzky
Keyword(s):  
Group Theory ◽  
Abelian Subgroup ◽  
Profinite Group ◽  
Profinite Groups ◽  
Profinite Completion ◽  
Maximal Abelian Subgroup ◽  
Free Profinite Group ◽  
Abelian Subgroups

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

On Nonabelian H2 for Profinite Groups

1992 ◽  
Vol 44 (2) ◽  
pp. 388-399
Author(s):  
K.-H. Ulbrich
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Profinite Group ◽  
Profinite Groups ◽  
Image Position

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

scholarly journals DERIVED SUBGROUPS OF FIXED POINTS IN PROFINITE GROUPS

2011 ◽  
Vol 54 (1) ◽  
pp. 97-105
Author(s):  
CRISTINA ACCIARRI ◽  
ALINE DE SOUZA LIMA ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Abelian Group ◽  
Fixed Points ◽  
Profinite Group ◽  
Elementary Abelian Group ◽  
Profinite Groups ◽  
Locally Finite ◽  
Group Of Automorphisms

AbstractThe main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that CG(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

Effective aspects of profinite groups

1981 ◽  
Vol 46 (4) ◽  
pp. 851-863 ◽  
Author(s):  
Rick L. Smith
Keyword(s):  
Finite Groups ◽  
Inverse Limit ◽  
Commutator Subgroup ◽  
Galois Groups ◽  
Profinite Group ◽  
Profinite Groups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Work Done ◽  
General Method

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized (effectively!) by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.The commutator subgroup, the Frattini subgroup, thep-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem.

scholarly journals Profinite groups with restricted centralizers of commutators

2019 ◽  
Vol 150 (5) ◽  
pp. 2301-2321 ◽  
Author(s):  
Eloisa Detomi ◽  
Marta Morigi ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Index ◽  
Profinite Group ◽  
Profinite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractA group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

scholarly journals Compact groups with many elements of bounded order

2020 ◽  
Vol 23 (6) ◽  
pp. 991-998
Author(s):  
Meisam Soleimani Malekan ◽  
Alireza Abdollahi ◽  
Mahdi Ebrahimi
Keyword(s):  
Group Theory ◽  
Haar Measure ◽  
Open Subgroup ◽  
Profinite Group ◽  
Compact Groups ◽  
Profinite Groups ◽  
Russian Academy Of Sciences ◽  
Kourovka Notebook ◽  
Commuting Pairs ◽  
Academy Of Sciences

AbstractLévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

scholarly journals The weight of nonstrongly complete profinite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamar Bar-On
Keyword(s):  
Profinite Group ◽  
Profinite Groups ◽  
Profinite Completion ◽  
Local Weight

Abstract We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

scholarly journals Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

2021 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Prime Order ◽  
Profinite Group ◽  
Nilpotent Subgroup ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Engel Element

AbstractA right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

Coverings and dimensions in infinite profinite groups

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Peter Maga
Keyword(s):  
Hausdorff Dimension ◽  
Compact Subset ◽  
Set Theory ◽  
Profinite Group ◽  
Box Dimension ◽  
Profinite Groups

AbstractAnswering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.

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