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.

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.

scholarly journals FITTING SUBGROUP AND NILPOTENT RESIDUAL OF FIXED POINTS

2018 ◽  
Vol 100 (1) ◽  
pp. 61-67
Author(s):  
EMERSON DE MELO ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Abelian Group ◽  
Fixed Points ◽  
Elementary Abelian Group ◽  
Fitting Subgroup ◽  
Nilpotent Residual ◽  
Formula Group

Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

A Sufficient Condition for Solvability in Groups Admitting Elementary Abelian Operator Groups

1977 ◽  
Vol 29 (4) ◽  
pp. 848-855 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Prime Divisor ◽  
Sylow Subgroup ◽  
Elementary Abelian Group ◽  
Sufficient Condition ◽  
Group Of Automorphisms ◽  
Critical Observation ◽  
A Finite Group

Generalizing a celebrated theorem of Thompson, R. P. Martineau has established [4; 5] that a finite group which admits an elementary abelian group of automorphisms with trivial fixed-point subgroup is necessarily solvable. A critical observation in his approach to this problem is the fact that, corresponding to each prime divisor of its order, such a group contains a unique Sylow subgroup invariant (as a set) under the action. Hence, the theorem we shall derive here represents a modest extension of Martineau's result.

scholarly journals WORDS AND PRONILPOTENT SUBGROUPS IN PROFINITE GROUPS

2014 ◽  
Vol 97 (3) ◽  
pp. 343-364 ◽  
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY
Keyword(s):  
Profinite Group ◽  
Profinite Groups ◽  
Locally Finite ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

scholarly journals Profinite groups and centralizers of coprime automorphisms whose elements are Engel

2018 ◽  
Vol 21 (3) ◽  
pp. 485-509
Author(s):  
Cristina Acciarri ◽  
Danilo Sanção da Silveira
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Elementary Abelian Group ◽  
Profinite Groups ◽  
Bounded Number ◽  
Formula Group ◽  
Quantitative Results

Abstract Let q be a prime, n a positive integer and A an elementary abelian group of order {q^{r}} with {r\geq 2} acting on a finite {q^{\prime}} -group G. We show that if all elements in {\gamma_{r-1}(C_{G}(a))} are n-Engel in G for any {a\in A^{\#}} , then {\gamma_{r-1}(G)} is k-Engel for some {\{n,q,r\}} -bounded number k, and if, for some integer d such that {2^{d}\leq r-1} , all elements in the dth derived group of {C_{G}(a)} are n-Engel in G for any {a\in A^{\#}} , then the dth derived group {G^{(d)}} is k-Engel for some {\{n,q,r\}} -bounded number k. Assuming {r\geq 3} , we prove that if all elements in {\gamma_{r-2}(C_{G}(a))} are n-Engel in {C_{G}(a)} for any {a\in A^{\#}} , then {\gamma_{r-2}(G)} is k-Engel for some {\{n,q,r\}} -bounded number k, and if, for some integer d such that {2^{d}\leq r-2} , all elements in the dth derived group of {C_{G}(a)} are n-Engel in {C_{G}(a)} for any {a\in A^{\#},} then the dth derived group {G^{(d)}} is k-Engel for some {\{n,q,r\}} -bounded number k. Analogous (non-quantitative) results for profinite groups are also obtained.

scholarly journals On profinite groups in which commutators are Engel

2001 ◽  
Vol 70 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Pavel Shumyatsky
Keyword(s):  
Finite Order ◽  
Profinite Group ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Additional Assumptions

AbstractWe show that if G is a finitely generated profinite group such that [x1, x2, …, xk] is Engel for any x1, x2, …, xk ∈ G, then γ(G) is locally nilpotent, and if [x1, x2, …, xk] has finite order for any x1, x2, …, xk ∈ G then, under some additional assumptions, γk(G) is locally finite.

scholarly journals CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

2019 ◽  
Vol 62 (1) ◽  
pp. 183-186
Author(s):  
KIVANÇ ERSOY
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Fixed Points ◽  
Simple Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Locally Finite Group ◽  
Simple Groups ◽  
Locally Finite ◽  
Locally Finite Groups

AbstractIn Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.

Functors on Finite Vector Spaces and Units in Abelian Group Rings

1986 ◽  
Vol 29 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group ◽  
Vector Spaces ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
Group Of Units ◽  
Finite Vector ◽  
Finite Vector Spaces

AbstractIf A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the compositewhere C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

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