共查询到20条相似文献,搜索用时 46 毫秒
1.
It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]). 相似文献
2.
Let G be an infinite finitely generated pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. We prove that G splits over an edge stabilizer either as an amalgamated free pro-p product or as a pro-p \({\text {HNN}}\)-extension. Using this result, we prove under a certain condition that free pro-p products with procyclic amalgamation inherit from its amalgamated free factors the property of each 2-generated pro-p subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogues of classical results in abstract combinatorial group theory. 相似文献
3.
Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology. 相似文献
4.
B. M. Veretennikov 《Proceedings of the Steklov Institute of Mathematics》2012,278(1):139-151
We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C n 2 for p≠ 3 and d(G′) ≤ C n 2 + C n 3 for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C n 2 . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C n 2 + C n 3 , then G″ is an elementary abelian group. 相似文献
5.
Ken Söze 《Israel Journal of Mathematics》2017,219(2):817-834
We show that for every finitely presented pro-p nilpotent-by-abelian-by-finite group G there is an upper bound on \({\dim _{{\mathbb{Q}_p}}}\left( {{H_1}\left( {M,{\mathbb{Z}_p}} \right){ \otimes _{{\mathbb{Z}_p}}}{\mathbb{Q}_p}} \right)\), as M runs through all pro-p subgroups of finite index in G. 相似文献
6.
V. A. Roman’kov N. G. Khisamiev A. A. Konyrkhanova 《Siberian Mathematical Journal》2017,58(3):536-545
We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT n (Z) has no proper existentially closed subgroups. 相似文献
7.
Bogomolov multipliers for some <Emphasis Type="Italic">p</Emphasis>-groups of nilpotency class 2
下载免费PDF全文
![点击此处可从《数学学报(英文版)》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Ivo Michailov 《数学学报(英文版)》2016,32(5):541-552
The Bogomolov multiplier B 0(G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether’s problem. We show that if G is a central product of G 1 and G 2, regarding K i ≤ Z(G i ), i = 1, 2, and θ: G 1 → G 2 is a group homomorphism such that its restriction \(\theta {|_{{K_1}}}:{K_1} \to {K_2}\) is an isomorphism, then the triviality of B 0(G 1/K 1),B 0(G 1) and B 0(G 2) implies the triviality of B 0(G). We give a positive answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity). 相似文献
8.
HaiPeng Qu 《中国科学 数学(英文版)》2010,53(11):3037-3040
A subgroup A of a p-group G is said to be soft in G if C G (A) = A and |N G (A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4. 相似文献
9.
A subgroup of index p k of a finite p-group G is called a k-maximal subgroup of G. Denote by d(G) the number of elements in a minimal generator-system of G and by δ k (G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G. In this paper, the authors classify the finite p-groups with δd(G)(G) ≤ p2 and δd(G)?1(G) = 0, respectively. 相似文献
10.
The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with Op(G) = 1 then |G/F(G)|p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite). 相似文献
11.
D. N. Azarov 《Russian Mathematics (Iz VUZ)》2017,61(5):1-6
Let p be a prime number. Recall that a group G is said to be a residually finite p-group if for every non-identity element a of G there exists a homomorphism of the group G onto a finite p-group such that the image of a does not coincide with the identity. We obtain a necessary and sufficient condition for the free product of two residually finite p-groups with finite amalgamated subgroup to be a residually finite p-group. This result is a generalization of Higman’s theorem on the free product of two finite p-groups with amalgamated subgroup. 相似文献
12.
A. G. Erschler 《Functional Analysis and Its Applications》2005,39(4):317-320
We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality v G,S (n) ? ρ(n) for all n. For every prime p, one can take G to be a p-group; one can also take a torsion-free group G. We also discuss some generalizations of this assertion. 相似文献
13.
In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FPn over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FPn but not of type FPn+1 over Z? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FPn fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler. 相似文献
14.
Lijian An 《Frontiers of Mathematics in China》2018,13(4):763-777
Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use pM(G) and pm(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G) < 2m(G)?1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are pk and pk+1. 相似文献
15.
A. Vera-López J. M. Arregi Leyre Ormaetxea F. J. Vera-López Olga Basova 《Israel Journal of Mathematics》2016,214(2):675-697
We denote by Gn the group of the upper unitriangular matrices over Fq, the finite field with q = pt elements, and r(Gn) the number of conjugacy classes of Gn. In this paper, we obtain the value of r(Gn) modulo (q2 -1)(q -1). We prove the following equalities 相似文献
16.
In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively. 相似文献
17.
For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K 0(R e p?G?S n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k t h -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n). 相似文献
18.
19.
Let Γg,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φf(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φf(G) is nilpotent when G is a finitely generated subgroup of Γg,b, in this paper we compute Φf(G) for certain subgroups of Γg,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b. 相似文献
20.
A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = xkfor all x ∈ G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed. 相似文献