arXiv subjects
Publications and source records attributed to Pavel Shumyatsky.
Let $G$ be a finite group admitting a coprime automorphism $α$. Let $J_G(α)$ denote the set of all commutators $[x,α]$, where $x$ belongs to an $α$-invariant Sylow subgroup of $G$. We show that $[G,α]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(α)$ is soluble or nilpotent, respectively.
For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p group, then G is virtually of rank at most r+1.
Let $G$ be a finite group. A coprime commutator in $G$ is any element that can be written as a commutator $[x,y]$ for suitable $x,y\in G$ such that $π(x)\capπ(y)=\emptyset$. Here $π(g)$ denotes the set of prime divisors of the order of the element $g\in G$. An anti-coprime commutator is an element that can be written as a commutator $[x,y]$, where $π(x)=π(y)$. The main results of the paper are as follows. -- If $|x^G|\leq n$ whenever $x$ is a coprime commutator, then $G$ has a nilpotent subgroup of $n$-bounded index. -- If $|x^G|\leq n$ for every anti-coprime commutator $x\in G$, then $G$ has a subgroup $H$ of nilpotency class at most $4$ such that $[G : H]$ and $|γ_4 (H)|$ are both $n$-bounded. We also consider finite groups in which the centralizers of coprime, or anti-coprime, commutators are of bounded order.
Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, if every pair of elements from $I_G(α)$ generates a soluble, or nilpotent, subgroup, then $[G,α]$ is soluble, or respectively nilpotent.
For a group G and an element a in G let |a|_k denote the cardinality of the set of commutators [a,x_1,...,x_k], where x_1,...,x_k range over G. The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n such that |x|_k < n for every x in G. More precisely, if |x|_k < n for every x in G then γ_{k+1}(G) has finite (k,n)-bounded order.
A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $π$, we take interest in profinite groups with restricted centralizers of $π$-elements. It is shown that such a profinite group has an open subgroup of the form $P\times Q$, where $P$ is an abelian pro-$π$ subgroup and $Q$ is a pro-$π'$ subgroup. This significantly strengthens a result from our earlier paper.
Given a group-word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. The word $w$ is concise if $w(G)$ is finite for all groups $G$ in which $G_w$ is finite. We obtain several results supporting the conjecture that the word $[u_1,\dots,u_s]$ is concise whenever the words $u_1,\dots,u_s$ are non-commutator.
Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group $G'$ is finite. We establish the following result. Let $n$ be a positive integer and $K$ a subgroup of a group $G$ such that $|x^G|\leq n$ for each $x\in K$. Let $H=\langle K^G\rangle$ be the normal closure of $K$. Then the order of the derived group $H'$ is finite and $n$-bounded. Some corollaries of this result are also discussed.
For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq ε> 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G : H] $ and $|γ_4(H)|$ are both bounded in terms of $ε$. We also show that if $G$ is an infinite group whose commutators have boundedly many conjugates, or indeed if $G$ satisfies a certain more general commutator covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.
We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$ only. In the case where $G$ is soluble we obtain a better result -- if $G$ is a finite soluble group in which every nilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of $w(G)$ is at most $r+1$.
Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geqε$ for some fixed $ε>0$. We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indexes $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $ε$-bounded. This extends the well known theorem, due to P. M. Neumann, that covers the case where $K=G$. We deduce a number of corollaries of this result. A typical application is that if $K$ is the generalized Fitting subgroup $F^*(G)$ then $G$ has a class-2-nilpotent normal subgroup $R$ such that both the index $[G:R]$ and the order of the commutator subgroup $[R,R]$ are $ε$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.
A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same primes. In the present paper we establish that a profinite group $G$ is finite-by-pronilpotent if the cardinality of the set of coprime commutators in $G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite commutator subgroup $G'$ if the cardinality of the set of anti-coprime commutators in $G$ is less than $2^{\aleph_0}$.
Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup $G'$ is finite. We establish the following results. Let $K,n$ be positive integers and $G$ a group having a $K$-approximate subgroup $A$. If $|a^G|\leq n$ for each $a\in A$, then the commutator subgroup of $\langle A^G\rangle$ has finite $(K,n)$-bounded order. If $|[g,a]^G|\leq n$ for all $g\in G$ and $a\in A$, then the commutator subgroup of $[G,A]$ has finite $(K,n)$-bounded order.
Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the cardinality of $G_w$ is less than $2^{\aleph_0}$ and $w(G)$ is generated by finitely many $w$-values. For several families of words $w$ we show that under these assumptions $w(G)$ must be finite. Our results are related to the concept of conciseness of group words.
Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $δ_k$-values $a,b\in G$ of coprime orders. In the course of the proof we establish the following result of independent interest: Let $P$ be a Sylow $p$-subgroup of $G$. Then $P\cap G^{(k)}$ is generated by $δ_k$-values contained in $P$. This is related to the so-called Focal Subgroup Theorem.
A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G$ in $\mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words -- and the particular words $x^2$ and $[x^2,y]$ -- have the property that the corresponding verbal subgroup is finite in a profinite group $G$ whenever the word takes at most countably many values in $G$. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if $w$ is one of the group words $x^2$, $x^3$, $x^6$, $[x^3,y]$ or $[x,y,y]$, then $w$ is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families $[x^m,z_1,\ldots,z_r]$ and $[x,y,y,z_1,\ldots,z_r]$, where $m \in \{2,3\}$ and $r \ge 1$.
The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of finite CP-groups yields a detailed description of profinite groups with that property. Then we deal with two generalizations of profinite CP-groups. Theorem 1.2. A profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x in G there is a prime p (depending on x) such that the centralizer of x is virtually pro-p. Theorem 1.3. Let G be a profinite group in which each element has either finite or prime power (possibly infinite) order. Then G is either torsion or virtually pro-p for some prime p.
The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.