170,505 research outputs found
Centralizers of coprime automorphisms of finite groups
No full textLet A be an elementary abelian group of order pk with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γk−2(CG(a)) is nilpotent of class at most c for any a ∈ A#, then γk−2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2d + 2 ≤ k, the dth derived group of CG (a) is nilpotent of class at most c for any a ∈ A#, then the dth derived group G(d) is nilpotent and has {c, k, p}-bounded
nilpotency class
Double automorphisms of graded Lie algebras
No full textWe introduce the concept of a double automorphism of an -graded Lie algebra . Roughly, this is an automorphism of which also induces an automorphism of the group . It is clear that the set of all double automorphisms of forms a subgroup in . In the present paper we prove several nilpotency criteria for a graded Lie algebra admitting a finite group of double automorphisms. One of the obtained results is as follows.
Let be a torsion-free abelian group and an -graded Lie algebra in which . Assume that admits a finite group of double automorphisms such that for all nontrivial and is nilpotent of class . Then is nilpotent and the class of is bounded in terms of , and only.
We also give an application of our results to groups admitting a Frobenius group of automorphisms
On the rank of a finite group of odd order with an involutory automorphism
No full textLet G be a finite group of odd order admitting an involutory automorphism φ, and let G-φ be the set of elements of G transformed by φ into their inverses. Note that [ G, φ] is precisely the subgroup generated by G-φ. Suppose that each subgroup generated by a subset of G-φ can be generated by at most r elements. We show that the rank of [ G, φ] is r-bounded
On groups in which Engel sinks are cyclic
No full textFor an element g of a group G, an Engel sink is a subset E(g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to E(g). We conjecture that if G is a profinite group in which every element admits a sink that is a procyclic subgroup, then G is procyclic-by-(locally nilpotent). We prove the conjecture in two cases – when G is a finite group, or a soluble pro-p group
Varieties of groups and the problem on conciseness of words
No full textA group-word is concise in a class of groups if and only if the verbal subgroup is finite whenever takes only finitely many values in a group . It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word in residually finite groups is equivalent to that in the class of virtually pro- groups.
This is used to show that if are positive integers and is a multilinear commutator word, then the words and are concise in residually finite groups. Earlier this was known only in the case where is a prime power.
In the course of the proof we establish that certain classes of groups satisfying the law , or , are varieties
On words that are concise in residually finite groups
No full textA group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G in the class X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word w^q is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γ_k the word w^q is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q is actually concise in the class of all groups
A stronger form of Neumann's BFC-theorem
No full textGiven 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| ≤ n for each x ∈ K. Let H=⟨K^G⟩ 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
On finite groups in which coprime commutators are covered by few cyclic subgroups
No full textThe coprime commutators γj∗ and δj∗ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. Every element of a finite group G is both a γ1∗-commutator and a δ0∗-commutator. Now let j>=2 and let X be the set of all elements of G that are powers of γj-1∗ -commutators. An element g is a γj∗ -commutator if there exist a ∈ X and b ∈ G such that g = [a,b] and (|a|,|b|) = 1. For j>=1 let Y be the set of all elements of G that are powers of δj-1∗ -commutators. An element g is a δj∗ -commutator if there exist a,b ∈ Y such that g = [a,b] and (|a|,|b|) = 1. The subgroups of G generated by all γj∗-commutators and all δj∗-commutators are denoted by γj∗(G) and δj∗(G), respectively. For every j >=2 the subgroup γj∗(G) is precisely the last term γ∞(G) of the lower central series of G, while for every j>=1 the subgroup δj∗(G) is precisely the last term of the lower central series of δj∗−1(G), that is, δj∗(G) = γ∞ (δj-1∗ (G)).
In the present paper we prove that if G possesses m cyclic subgroups whose union contains all γj∗-commutators of G, then γj∗(G) contains a subgroup Δ, of m-bounded order, which is normal in G and has the property that γj∗(G)/Δ is cyclic. If j>=2 and G possesses m cyclic subgroups whose union contains all δj∗-commutators of G, then the order of δj∗(G) is m-bounded
Genus, thickness and crossing number of graphs encoding the generating properties of finite groups
No full textAssume that G is a finite group and let a and b be non-negative integers. We define an undirected graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,...,xa) and (y1,...,yb) are adjacent if and only if 〈x1,...,xa,y1,...,yb〉=G. Our aim is to estimate the genus, the thickness and the crossing number of the graph Γa,b(G) when a and b are positive integers, giving explicit lower bounds on these invariants in terms of |G|
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