scholarly journals Weight Parameterization of Simple Modules for p-Solvable Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 1-46
Author(s):  
Lluis Puig
Keyword(s):  
Solvable Groups ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Outer Automorphisms ◽  
Odd Order ◽  
The One ◽  
A Finite Group ◽  
The Relationship

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

scholarly journals Parameterization of Irreducible Characters for p-Solvable Groups

2012 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Irreducible Characters ◽  
Outer Automorphisms ◽  
A Finite Group

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

THE NUMBER OF CONJUGACY CLASSES CONTAINED IN NORMAL SUBGROUPS OF SOLVABLE GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250204
Author(s):  
AMIN SAEIDI ◽  
SEIRAN ZANDI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group and let N be a normal subgroup of G. Assume that N is the union of ξ(N) distinct conjugacy classes of G. In this paper, we classify solvable groups G in which the set [Formula: see text] has at most three elements. We also compute the set [Formula: see text] in most cases.

scholarly journals THE 2-RANKS OF HYPERELLIPTIC CURVES WITH EXTRA AUTOMORPHISMS

2009 ◽  
Vol 05 (05) ◽  
pp. 897-910 ◽  
Author(s):  
DARREN GLASS
Keyword(s):  
Automorphism Group ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Closed Field ◽  
Base Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Carry Over ◽  
The Relationship

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

Weak Cayley tables and generalized centralizer rings of finite groups

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
Author(s):  
STEPHEN P. HUMPHRIES ◽  
EMMA L. RODE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Cayley Table ◽  
A Finite Group ◽  
Finite Conjugacy ◽  
Relationship Of ◽  
Cayley Tables ◽  
The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

Solution of Brauer’s k ⁢ ( B ) k(B) -conjecture for π-blocks of π-separable groups

2018 ◽  
Vol 30 (4) ◽  
pp. 1061-1064
Author(s):  
Benjamin Sambale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Semidirect Product ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
The Other ◽  
Other Hand ◽  
A Finite Group

Abstract Answering a question of Pálfy and Pyber, we first prove the following extension of the {k(GV)} -problem: Let G be a finite group and let A be a coprime automorphism group of G. Then the number of conjugacy classes of the semidirect product {G\rtimes A} is at most {\lvert G\rvert} . As a consequence, we verify Brauer’s {k(B)} -conjecture for π-blocks of π-separable groups which was proposed by Y. Liu. This generalizes the corresponding result for blocks of p-solvable groups. We also discuss equality in Brauer’s Conjecture. On the other hand, we construct a counterexample to a version of Olsson’s Conjecture for π-blocks which was also introduced by Liu.

On class numbers of a finite group and of its subgroups

1993 ◽  
Vol 123 (2) ◽  
pp. 295-301
Author(s):  
A. Vera-López ◽  
J. Sangroniz
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Class Numbers ◽  
Normal Subgroups ◽  
A Finite Group ◽  
The Relationship

SynopsisIn this paper we obtain new results which relate the number of conjugacy classes of л-elements of a finite group and an arbitrary subgroup, which are analogous to some results about normal subgroups. We also prove some new results which show the relationship between class numbers and splitting theorems. Our proofs only involve elementary techniques.

Groups with few conjugacy classes

2011 ◽  
Vol 54 (2) ◽  
pp. 423-430 ◽  
Author(s):  
László Héthelyi ◽  
Erzsébet Horváth ◽  
Thomas Michael Keller ◽  
Attila Maróti
Keyword(s):  
Finite Group ◽  
Prime Divisor ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
A Finite Group

AbstractLet G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have $k(G)\geq2\smash{\sqrt{p-1}}$ with equality if and only if if $\smash{\sqrt{p-1}}$ is an integer, $G=C_{p}\rtimes\smash{C_{\sqrt{p-1}}}$ and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller.

A Note on the p-Deficiency Class of a Finite Group

2012 ◽  
Vol 19 (02) ◽  
pp. 353-358 ◽  
Author(s):  
Tianze Li ◽  
Weigang Xu ◽  
Jiping Zhang
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Characteristic P ◽  
P Deficiency ◽  
Class 1 ◽  
A Finite Group ◽  
P Type ◽  
The Relationship

In this note, we explore the relationship between finite groups of characteristic p type and those of p-deficiency class 1. We study the structure of finite groups of characteristic p type. Besides, we show that the p-rank (resp., p-length) of a p-solvable group which is of exact p-deficiency class r(> 0) is bounded by r (resp., a function of r).

scholarly journals LANDAU’S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS

2016 ◽  
Vol 104 (1) ◽  
pp. 37-43
Author(s):  
MARK L. LEWIS
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Character Theory ◽  
Finite Solvable Group ◽  
Roots Of Unity ◽  
Prime Power Order ◽  
A Finite Group

When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

scholarly journals Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

2014 ◽  
Vol 66 (6) ◽  
pp. 1201-1224 ◽  
Author(s):  
Jeffrey D. Adler ◽  
Joshua M. Lansky
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Point Group ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Reductive Groups ◽  
A Finite Group ◽  
The Relationship

AbstractSuppose that is a connected reductive group defined over a field k, and ┌ is a finite group acting via k-automorphisms of satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of -fixed points in is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair (,┌) and consider any group G satisfying the axioms. If both and G are k-quasisplit, then we can consider their duals *and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G*(k) to the analogous set for *(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

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