The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .     

Fitting Height and Diameter of Character Degree Graphs

Let a graph Γ have bounded Fitting height (i.e., there is a bound on the Fitting heights of those groups whose character degree graph is Γ) and G be any solvable group with character degree graph Γ and Fitting height h(G). We improve Moretò's bound by proving that if no vertex in Γ is adjacent to every other one, then h(G) ≤4, else h(G) ≤6. As a consequence, if a solvable group G has character degree graph with diameter 3, then h(G) ≤4. Moreover, G has at most one non-abelian normal Sylow subgroup in this case.     

Groups with free 2-generator subsemigroups

A torsion-free group G is termed an X-group if the subsemigroup generated by any pair of noncommuting elements of G contains a free semigroup of rank two. Similarly a torsion-free group G is termed a Y-group if either the subsemigroup generated by x and y is free or else the subsemigroup generated by x and y⁻¹ is free whenever x and y are noncommuting elements of G. Our main result is that if A and T are respectively both X-groups or both Y-groups, then their standard wreath product A ∝ T is an X-group or a Y-group. It follows that free solvable groups are X-groups and also Y-groups, which generalizes earlier work of Mal'cev. Support from the National Science Foundation, the SRC and the hospitality of Cambridge University is gratefully acknowledged.     

CC-groups with periodic central factor

A group G is said to be a group with Černikov conjugacy classes or a CC-group if it induces on the normal closure of each one of its elements a group of automorphisms which is a Černikov group, that is, a finite extension of an abelian group satisfying the minimal condition on subgroups. This concept is a natural extension of that an FC-group, that is, a group in which every element has a finite number of conjugates. It is known that if G is an FC-group then the central factor G/Z(G) is periodic. This result does not hold for CC-groups and in this paper we study CC-groups G in which the central factor G/Z(G) is periodic, a finiteness condition which has a deep influence on the structure of the group G. In particular, we characterize those CC-groups as above that are FC-groups by imposing some additional conditions on their structure. This research has been supported by DGICYT (Spain) PS88-0085     

External Paradoxical Decompositions

Ore's condition states that a cancellative semigroup S which has common right multiples embeds into a group G such that certain properties are satisfied by S and G. We show that G is nonamenable if and only if the semigroup S^-1 is G-paradoxical with respect to right multiplication by elements of S. We explore certain properties of this decomposition of S^-1.     

ESSENTIALLY FINITELY INDECOMPOSABLE ABELIAN p-GROUPS

Abstract

An abelian p-group C is said to be essentially finitely indecomposable (efi) if given any decomposition of G as the direct sum of a family of subgroups, there exists a positive integer n such that all but at moat a finite number of subgroups of this family are bounded by n. We look at examples and related questions. We prove that a reduced abelian p-group G is efi if and only if G modulo its elements of infinite height is efi. In the proof of this we obtain the following result which is of independent interest: Let A be a reduced p-group with a summand K such that K is a direct sum of cyclic groups. Let B be a basic subgroup of A. Then B contains a subgroup C such that C is a summand of A and the final rank of C is equal to the final rank of K.     

Semisimple crossed products of groups and rings

Let K?G be a crossed product of a multiplicative group G over an associative ring K with 1 and let C(G) be the center of G. If K has no C(G)-invariant ideals, then the Jacobson radical of the center of K?G is a nil ideal. In addition, if G is a ZA-group, then K?G is semisimple if and only if K?G has no central nilpotent elements.     

Inverse Near Permutation Semigroups

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group G of permutations on N elements and by a set H of transformations of rank N - 1. In this paper we give necessary and sufficient conditions for a near permutation semigroup S = ‹G,H›, where H is a group, to be inverse. Moreover, we obtain conditions which guarantee that its semilattice of idempotents is generated by the idempotents of S of rank greater than N - 2 or N - 3.     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

Lambda number of the power graph of a finite group

Journal of Algebraic Combinatorics - The power graph $$\Gamma _G$$ of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of...     

Galois subrings of simple rings

Suppose R is a finite direct sum of simple associative rings and G is a finite group of auto-morphisms of the ring R. It is shown that if there is no additive ¦G¦-torsion in R, then the subring of elements of R that are fixed under G is a finite direct sum of simple rings.     

A Class of Solvable Lie Algebras with Triangular Decompositions

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K^# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K^# such that all elements of K^# belonging to the connected component C of K^# containing x are at most distance 2 from c and N^G(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime divisor of the order of the complement, generates a solvable subgroup together with any ofits conjugates. In the third section we prove a generalization of the Glauberman-Thompson normal p-complement theorem, and using this wegive another sufficient condition for the Frobenius kernel to be a normal subgroup for |G| odd, namely we prove this under the conditionthat all the Sylow normalizers in G intersect some of the complements     

The structure of blocks

In this paper, we focus on the structure of p-blocks with defect group satisfying some special condition. These special conditions include: two elements of the defect group are conjugate to each other in defect D if and only if they are conjugate to each other in G; the number of conjugacy classes whose p-part is contained in P by conjugacy is not larger than ∣P∣     

Nilpotent elements in group rings

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.This work has been supported by N.R.C. Grant No. A-5300.     

On locally graded groups with an Engel condition on infinite subsets

A group G is said to be in E_k^*E_k^* (k a positive integer), if every infinite subset of G contains a pair of elements that generate a k-Engel group.¶It is shown that a finitely generated locally graded group G in E_k^*E_k^* is a finite-by- (k-Engel) group, in particular a finite extension of a k-Engel group.     

分次环与局部化

设Ｇ是有限群，Ｒ是有单位元的Ｇ－型分次环，Ｓ是包含在Ｒ的所有齐次元素组成的集合内的乘法封闭子集，Ｓ＝ｘ∈Ｇａｅ（ｇｘ，ｘ）ａ∈Ｓ，Ｄｅｇ（ａ）＝ｇ∈Ｇ｛｝，Ｓ＝＝ｘ∈Ｇａｅ（ｇｘ，ｘｈ）ａ∈Ｓ，Ｄｅｇ（ａ）＝ｇ∈Ｇ，ｈ∈Ｇ｛｝，ＭＧ（Ｒ）表示以Ｇ的元作为行列标的｜Ｇ｜阶矩阵环．本文证明了Ｒ关于Ｓ满足左Ｏｒｅ条件当且仅当Ｒ＃Ｇ关于Ｓ满足左Ｏｒｅ条件当且仅当ＭＧ（Ｒ）关于Ｓ＝满足左Ｏｒｅ条件，而且，Ｓ－１（Ｒ＃Ｇ）≌（Ｓ－１Ｒ）＃Ｇ和Ｓ＝，－１（ＭＧ（Ｒ））≌ＭＧ（Ｓ－１Ｒ）．     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers     

Characterization of L2(16) by Τe(L2(16))

Let G be a group and πe(G) the set of element orders of G.Let k∈πe(G) and m k be the number of elements of order k in G.Letτe(G)={mk|k∈πe(G)}.In this paper,we prove that L2(16) is recognizable byτe (L2(16)).In other words,we prove that if G is a group such that τe(G)=τe(L2(16))={1,255,272,544,1088,1920},then G is isomorphic to L2(16).     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.