A Generalized Brauer Induction Theorem and Its Converse

2012 ◽  
Vol 19 (03) ◽  
pp. 427-432 ◽  
Author(s):  
Gang Chen
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Complex Number ◽  
Prime Numbers ◽  
Linear Map ◽  
The Family ◽  
A Finite Group ◽  
Complex Number Field ◽  
Character Ring

Let G be a finite group, S a unitary subring of the complex number field ℂ and R(G) the character ring of G. Let π be the set of rational prime numbers whose inverses do not belong to S. Denote the family of all p-elementary subgroups of G by W(π), where p runs over π. It is proved that, in the sense of conjugation, W(π) is the least family [Formula: see text] of subgroups of G such that the S-linear map [Formula: see text] is surjective.

The Automorphisms of an Algebraically Closed Field

1970 ◽  
Vol 13 (1) ◽  
pp. 95-97 ◽  
Author(s):  
A. Charnow
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Closed Field ◽  
Complex Field ◽  
Algebraically Closed Field ◽  
The Family ◽  
Transcendency Degree ◽  
Complex Number Field

It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

1998 ◽  
Vol 41 (3) ◽  
pp. 267-278 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Linear System ◽  
Number Field ◽  
Complex Number ◽  
Complex Number Field

AbstractLet (X, L) be a polarized manifold over the complex number field with dim X = n. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if n = 3 and h0(L) ≥ 2, or dim Bs |L| ≤ 0 for any n ≥ 3. Moreover we can generalize the result of Sommese.

Phased Bessel functions

2008 ◽  
Vol 86 (7) ◽  
pp. 863-870 ◽  
Author(s):  
X Hu ◽  
H Wang ◽  
D -S Guo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Complex Number ◽  
Bessel Functions ◽  
Natural Extension ◽  
State Transitions ◽  
Photon State ◽  
Analytical Extension ◽  
Complex Number Field ◽  
Real Number Field

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

scholarly journals On characters in the principal 2-block

1978 ◽  
Vol 25 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Thomas R. Berger ◽  
Marcel Herzog
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cyclic Group ◽  
Complex Number ◽  
Irreducible Character ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

On the Schur-Zassenhaus theorem for groups of finite Morley rank

1992 ◽  
Vol 57 (4) ◽  
pp. 1469-1477 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Hall Subgroup ◽  
Morley Rank ◽  
Finite Group Theory ◽  
Hall Subgroups ◽  
Finite Morley Rank ◽  
A Finite Group

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact 1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group, where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p-Sylow for any prime p, or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H. Since ℂ* is abelian, these complements cannot be conjugated to each other.

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

Sign in / Sign up

Export Citation Format

Share Document