Sylow Products and the Solvable Residual

We study the connection between products of Sylow subgroups of a finite group G and the solvable residual of G. Let Π(𝒫) be a product of Sylow subgroups of G such that each prime divisor of |G| is represented exactly once in Π(𝒫). We prove that there exists a unique normal subgroup N of G which is minimal subject to the requirement Π(𝒫) N = G. Furthermore, N is perfect, and the product of all of these subgroups is the solvable residual of G. We also prove that the solvable residual of G is generated by all elements which arise from non-trivial factorizations of 1 _G in such products of Sylow subgroups.     

The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P ₁,…, P _m , where P _i is a Sylow p _i -subgroup of G, and g ∈ G, denote by m _𝒫(g) the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . Previously, it was shown that the properly normalized average of m _𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m _𝒫(gx) = m _𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.     

特征标次数的重数与可解群结构

非线性不可约特征标次数的重数全部为1的有限群的分类是熟知的.对可解群,本文讨论更一般的,即非线性不可约特征标次数的重数都与群阶互素的有限群的纯群论性质.特别地,得到了非线性不可约特征标次数的重数均小于2p的奇阶群G的分类结果.这里p为群阶|G|的最小素因子.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Normal Sylow subgroups and monomial Brauer characters

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G : ker φ|p' = φ(1)p' for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p' is the p'-part of φ(1).     

The Sylow p-Subgroups of a Limiting (Stable) Linear Group over an Algebraically Closed Field

In Math. Zh. 36(3) (1963), 240–246, we considered the classification of Sylow p-subgroups of a countable (finitary) symmetric group. Later, we extended this classification to Sylow p-subgroups of a limiting (stable) linear group over a finite field GF(q) with char(GF(q)) p. In this paper, the classification of the Sylow p-subgroups of the limiting (stable) linear group GL() is given for an algebraically closed field of characteristic p. Every Sylow p-subgroup is characterized by a p-adic integer together with a finite or countable cardinal.     

Solvable Permutation Groups and Orbits on Power Sets

We consider the class ? of finitely generated toral relatively hyperbolic groups. We show that groups from ? are commutative transitive and generalize a theorem proved by Benjamin Baumslag in [3 Baumslag, B. (1967). Residually free groups. Prceedings of the London Mathematical Society 17(3):402–418.[Crossref] , [Google Scholar]] to this class. We also discuss two definitions of (fully) residually-𝒞 groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii [18 Ol'shanskii, A. Yu. (1993). On residualing homomorphisms and G-subgroups of hyperbolic groups. International Journal of Algebra Computation 3:365–409.[Crossref] , [Google Scholar]] and Osin [22 Osin, D. V. (2010). Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of mathematics 172:1–39.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the equivalence of the two definitions for 𝒞 = ?. This is a generalization of the similar result obtained by Ol'shanskii for 𝒞 being the class of torsion-free hyperbolic groups. Let Γ ∈ ? be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14 Kharlampovich, O., Myasnikov, A. (2012). Limits of relatively hyperbolic groups and Lyndon's completions. Journal of the European Math. Soc. 14:659–680.[Crossref], [Web of Science ®] , [Google Scholar]] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from ?. On the other hand, we give an example of a finitely generated torsion-free fully residually-? group that does not embed into a group from ?; ? is the class of hyperbolic groups.     

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.     

Complete precones on noncommutative integral domains

A subgroup H is called Q-supplemented in a finite group G, if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _QG , where H _QG is the maximal quasinormal subgroup of G contained in H. In this article, we investigate the influence of Q-supplementation of some primary subgroups in finite groups. Some recent results are generalized.     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

On the Surjectivity of the Exponential Function of Solvable Lie Groups

For a solvable Lie group G the surjectivity of the exponential function exp_G is equivalent to the connectedness of the near-Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ? G there is an x ? g with g = exp_G x in which exp_G is regular.     

与Sylow-子群X-可置换的子群对有限群的影响

设X是有限群G中的一个非空子集,H和T是G的两个子群.称日与T在G中是X-可置换的,如果存在元素x∈X,满足HT~x=T~xH.作者探讨了当有限群G的某些子群与G的某些Sylow子群是X-可置换时G的结构.     

弱 c- 可置换子群对有限群结构的影响

Guo和Chen在2010年引入了弱c-可置换子群的概念,并利用这种新的广义置换性质对有限群的结构给出了一些新的刻画.本文推广了上述文献中的两个主要结论,确定了极小子群为弱c-可置换子群的有限群的结构.     

Bounding the Derived Length of a Solvable Group: An Improvement on a Result by Gluck

Let cd(G) be the set of degrees of irreducible complex characters and dl(G) be the derived length of the finite group G. It is a result by Gluck that if G is solvable then dl(G) ≤2|cd(G)|. Using a result of Isaacs and Knutson, I will in this article show that this bound can be improved to dl(G) ≤2|cd(G)| ?3 when |cd(G)| ≥3.     

On the Non-existence of Semiregular Relative Difference Sets in Groups with All Sylow Subgroups Cyclic

We show that a group with all Sylow subgroups cyclic (other than ) cannot contain a normal semiregular relative difference set (RDSs). We also give a new proof that dihedral groups cannot contain (normal) semiregular RDSs either.