scholarly journals A sharp upper bound for the size of Lusztig series

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Bessenrodt ◽  
Alexandre Zalesski
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
Simple Formula ◽  
Character Theory ◽  
Classical Groups ◽  
Irreducible Characters ◽  
Maximal Size ◽  
Groups Of Lie Type

AbstractThe paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group 𝐺 of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group 𝐺 in terms of its Lie rank and defining characteristic. When 𝐺 is specified as G(q) and 𝑞 is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of 𝐺.

scholarly journals Character Theory of Finite Groups with Trivial Intersection Subsets

1966 ◽  
Vol 27 (2) ◽  
pp. 515-524 ◽  
Author(s):  
John H. Walter
Keyword(s):  
Finite Groups ◽  
Forthcoming Paper ◽  
Character Theory ◽  
Irreducible Characters ◽  
Decomposition Numbers ◽  
Trivial Intersection ◽  
Effective Use

This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.

scholarly journals Gelfand–Graev Characters of the Finite Unitary Groups

10.37236/235 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
C. Ryan Vinroot
Keyword(s):  
Finite Groups ◽  
Unitary Group ◽  
Symmetric Functions ◽  
Point Of View ◽  
Unitary Groups ◽  
Character Theory ◽  
Characteristic Map ◽  
Finite Unitary Groups ◽  
Combinatorial Point ◽  
Groups Of Lie Type

Gelfand–Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand–Graev characters of the finite unitary group from a combinatorial point of view. In particular, we give the values of Gelfand–Graev characters at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand–Graev characters in terms of tableau combinatorics, and conclude with some multiplicity consequences.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

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