The Real Part of the Character Table of a Finite Group

In the ordinary character table of a finite group G, the values of the real valued irreducible characters on the real conjugacy classes form a sub-table which is square by Brauer's permutation lemma. We call this table the real part of the character table of G. Unlike the ordinary character table, viewed as a square matrix the real part of the character table is often singular. We present some results linking nonsingularity of this table to other properties of G.     

Recovering an Erased Row or Column in a Table of Characters for a Finite Group

We argue that an erased row or column in a table of characters of a finite group can be constructively recovered from retained data.     

自同构群是循环群被交换群扩张的有限群

设Ｃ是有限群，ＡｕｔＧ＝ＡＢ，，Ａ是交换群且每Ｓｙｌｏｗ子群的秩≤２，Ｂ是循环群，本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤２的交换群时，Ｇ循环。     

自同构群

设Ｇ是有限群，ＡｕｔＧ＝ＡＢ，Ｂ△ＡｕｔＧ，（｜Ａ｜，｜Ｂ｜）＝１，Ａ是交换群且每Ｓｙｏｗ子群的秩≤２，Ｂ是循环群。本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤的交换群时，Ｇ循环。     

Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras

We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra $\mathcal{M}$ into the *-algebra of measurable operators $\tilde {\mathcal {M}}$ endowed with the topology of convergence in measure is continuous. Using this fact, we prove that the topology of convergence in measure is a minimal one among all metrizable topologies consistent with the ring structure on $\tilde {\mathcal {M}}$ .     

Bounding an Index by the Largest Character Degree of a p-Solvable Group

In this article, we show that if p is a prime and G is a p-solvable group, then |G: O _p (G)| _p  ≤ (b(G) ^p /p)^1/(p?1), where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G: O _p (G)| _p  ≤ b(G).     

容许几乎单群的无平方因子阶2弧正则图

This paper completes the classification of (G, 2)-arc-regular graphs of square-free order where G is an almost simple group.     

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M?F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C _F (H) = 1 or |C _F (H)| =p and C _{F/C F (H)}(H) = 1. Suppose that M acts on a finite group G in such a manner that C _G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C _G (H))+ 1. Moreover, the Fitting series of C _G (H) coincides with the intersection of C _G (H) with the Fitting series of G.     

Character Values of Finite Groups

The aim of this paper is to describe simple methods to distribute the ordinary irreducible characters of finite groups into p-blocks and calculate the other character values from part of the values of the ordinary irreducible characters for p-blocks.2000 Mathematics Subject Classification: 20C20     

有限群的X-s-置换子群

Let X be a nonempty subset of a group G.A subgroup H of G is said to be X-s-permutable in G if there exists an element x ∈ X such that HPx = PxH for every Sylow subgroup P of G.In this paper,some new results are given under the assumption that some suited subgroups of G are X-s-permutable in G.     

The Existence of p-blocksof a Finite Group With a Given Defect Group

Brauer's problem 19［B］——in terms of group theoreticalproperties count the number of p-blocks with a given defect group——has been studied by many authors.Among them G. R. Robinson gave a pricise formula for the number. Of courseit is by no means easy to calculate by the formula. So it is still interesting tofind other formula counting the number or conditions for the existence ofp-blocks.     

Finite Groups with a Given Set of Character Degrees

Let G be a finite group with {1, q, r, qm, q ² m, rqm} as the character degree set, where r and q are distinct primes and m>1 is an integer not divisible by q or r. We show that G is solvable and the derived length of G equals 3.     

THE CHARACTER TABLE OF SYMMETRIC GROUPS

This paper gives two methods to construct simultaneously the character tables of thesymmetric groups■_(n-1),■_n,and ■_(n+1)with the aid of the character table of ■_(n-2)as well asthe author's formulae cited in[9].These methods require merely the rational operation ofintegers.     

The Existence of p-blocks of a Finite Group With a Given Defect Group

Brauer's problem 19[B] -- in terms of group theoretical properties count the number of p-blocks with a given defect group -- has been studied by many authors. Among them G. R.Robinson gave a pricise formula for the number. of course it is by no means easy to calculateby the formula. So it is still interesting to find other formula counting the number or conditionsfor the existence of p-blocks. In [St], for an element g of a finite group G, the author defined a(g,p)-vector of G (in this pape…     

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C _{C G (F)}(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.     

The Group of Automorphisms of the Rational Group Algebra of a Finite Metacyclic Group

We investigate the group of automorphisms Aut(? G) of a rational group algebra ? G of a finite metacyclic group G by first describing the simple components of the Wedderburn decomposition of ? G and then investigating when two of these simple components are isomorphic.