Search results for "General Mathematics"

Nilpotent-like fitting formations of finite soluble groups

2000

[EN] In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

A submatrix of the character table

2000

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

Surfaces non-orientables de genre deux

1993

The existence of nonorientable complete minimal surface of genus two, one end and total curvature −2π(2n+3),n≥3 is proved in this paper.

Uniform properties of collections of convex bodies

1991

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

1999

It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter-type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths, and where left and right lowest common multiples exist. A Garside monoid is a Gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition. A Gaussian group is the group of fractions of a Gaussian monoid, and a Garside group is the group of fractions of a Garside monoid. Braid groups and, more genera…

A Note on a Conjecture of Duval and Sturmian Words

2002

We prove a long standing conjecture of Duval in the special case of Sturmian words. Mathematics Subject Classication. ??????????????. Let U be a nonempty word on a nite alphabet A: A nonempty word B dierent from U is called a border of U if B is both a prex and sux of U: We say U is bordered if U admits a border, otherwise U is said to be unbordered. For example, U = 011001011 is bordered by the factor 011; while 00010001001 is unbordered. An integer 1 k n is a period of a word U = U1 :::U n if and only if for all 1 i n k we have Ui = Ui+k. It is easy to see that k is a period of U if and only if the prex B of U of length n k is a border of U or is empty. Let (U) denote the smallest period …

On the identities of the Grassmann algebras in characteristicp>0

2001

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM 2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM 2(K) over an infinite fieldK of positive odd characteristic, and to conjecture ba…

On the number of conjugacy classes of zeros of characters

2004

Letm be a fixed non-negative integer. In this work we try to answer the following question: What can be said about a (finite) groupG if all of its irreducible (complex) characters vanish on at mostm conjugacy classes? The classical result of Burnside about zeros of characters says thatG is abelian ifm=0, so it is reasonable to expect that the structure ofG will somehow reflect the fact that the irreducible characters vanish on a bounded number of classes. The same question can also be posed under the weaker hypothesis thatsome irreducible character ofG hasm classes of zeros. For nilpotent groups we shall prove that the order is bounded by a function ofm in the first case but only the derive…

Periodic and Nil Polynomials in Rings

1980

Let R be an associative ring and f(x1,…, xd) a polynomial in noncommuting variables. We say that f is periodic or nil in R if for all r1,…, rd ∈ R we have that f(r1,…, rd) is periodic, respectively nilpotent (recall that a ∈ R is periodic if for some integer ).

A non-linear Bishop–Phelps–BollobÁs type theorem

2018