1,721,123 research outputs found
On the minimal number of generators of finite non-abelian p-groups having an abelian automorphism group
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On finite-by-nilpotent profinite groups
Full text linkLet γn = [x1, ..., xn] be the nth lower central word. Suppose that G is a profinite group where the conjugacy classes xγn(G) contains less than 2א0 elements for any x ∈ G. We prove that then γn+1 (G) has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group G is finite-by-nilpotent if and only if there is a positive integer n such that xγn(G) contains less than 2א0 elements, for any x ∈ G
Profinite groups with restricted centralizers of commutators
Full text linkA group G has restricted centralizers if for each g in G the centralizer 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
Strong conciseness of coprime and anti-coprime commutators
Full text linkA 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 2א0. 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 2א0
Bounding the order of a verbal subgroup in a residually finite group
Full text linkLet w be a group-word. Given a group G, we denote by w(G) the verbal subgroup corresponding to the word w, that is, the subgroup generated by the set Gw of all w-values in G. The word w is called concise in a class of groups X if w(G) is finite whenever Gw is finite for a group G ∈χ. It is a long-standing problem whether every word is concise in the class of residually finite groups. In this paper we examine several families of group-words and show that all words in those families are concise in residually finite groups
Primitive permutation IBIS groups
Full text linkLet G be a finite permutation group on Ω. An ordered sequence of elements of Ω, (ω1,...,ωt), is an irredundant base for G if the pointwise stabilizer G(ωjavax.xml.bind.JAXBElement@606ca359,...,ωjavax.xml.bind.JAXBElement@77e0dd48) is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same size we say that G is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to PSL(2,2f)×PSL(2,2f) for some f≥2, in its diagonal action of degree 2f(22f−1)
Words of Engel type are concise in residually finite groups. Part II
Full text linkThis work is a natural follow-up of the article [5]. Given a group-word w and a group G, the verbal subgroup w.G/ is the one generated by all w-values in G. The word w is called concise if w.G/ is finite whenever the set of w-values in G is finite. It is an open question whether every word is concise in residually finite groups. Let w D w.x1; : : : ; xk/ be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word OEwq; ny is concise in residually finite groups (Theorem 1.2) while the word OEw; ny is boundedly concise in residually finite groups (Theorem 1.1)
On the rank of a verbal subgroup of a finite group
Full text linkWe 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 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 is at most
Finite groups with small centralizers of word-values
Full text linkGiven a positive integer m and a group-word w, we consider a finite group G such that w(G) ≠ 1 and all centralizers of non-trivial w-values have order at most m. We prove that if w=v(x1q1,⋯,xkqk), where v is a multilinear commutator word and q1, ⋯ , qk are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word w= [xn, y1, ⋯ , yk] with k≥ 1
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