102,770 research outputs found
POSITIVE LAWS ON LARGE SETS OF GENERATORS: COUNTEREXAMPLES FOR INFINITELY GENERATED GROUPS
No full textAbstractShumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes p, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.</jats:p
A focal subgroup theorem for outer commutator words
No full textLet be a finite group of order , where is a prime and is not divisible by , and let be a Sylow -subgroup of . If is an outer commutator word, we prove that is generated by the intersection of with the set of th powers of all values of in $G
W.H.O.L.E. : World Health Organization Learning Experiment (a W.H.O. game on development)
No full textOn 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
Graphs encoding the generating properties of a finite group
No full textAssume that G is a finite group. For every a, b ∈ N, we define a graph Γa,b(G) whose vertices correspond
to the elements of G a ∪ G b and in which two tuples (x1, . . . , xa) and (y1, . . . , yb) are adjacent if and only if
hx1, . . . , xa, y1, . . . , ybi = G. We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the
group structure, with the aim of understanding which information about G is encoded by these graphs
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
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
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 profinite groups in which commutators are covered by finitely many subgroups
No full textFor a family of group words w we show that if G is a profinite group in which all w-values are contained in a union of finitely many subgroups with a prescribed property, then the verbal subgroup w(G) has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If G contains finitely many subgroups G1 , G2 , . . . , Gs of finite exponent e whose union contains all γk-values in G, it is shown that γk(G) has finite (e,k,s)-bounded exponent. If G contains finitely many subgroups G1, G2, . . . , Gs of finite rank r whose union contains all γk-values, it is shown that γk(G) has finite (k,r,s)-bounded rank
Profinite groups with restricted centralizers of π -elements
No full textA group G is said to have restricted centralizers if for each g in G the centralizer CG(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 × 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
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