One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material about general linear algebraic groups before one c...

Oncogenic microRNAs (miRs) have emerged as diagnostic biomarkers and novel molecular targets for anti-cancer drug therapies. Real-time quantitative PCR (qPCR) is one of the most powerful techniques for analyzing miRs; however, the use of unsuitable normalizers might bias the results. Tumour heterogeneity makes even more difficult the selection of an adequate endogenous normalizer control. Here,...

Abstract If p and q are primes, G is a -solvable finite group, it possible to detect that -Sylow normalizer contained in using the character table of . This characterized terms degrees -Brauer characters. Some consequences, which include yet another generalization Itô–Michler theorem, also obtained.

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

Given a planar differential system with a first integral, we show how to find a normalizer. For systems with a center, we give an integral formula for the derivative of its period function. 0

1 Quaternions and the three-dimensional sphere 3 1.1 Hamiltion’s quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Lie group S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 The normalizer of Q8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 The exponential map for S . . . . . . . . . . . . . . . . ....

Let (AS , S) be an Artin-Tits and X a subset of S ; denote by AX the subgroup of AS generated by X. When AS is of spherical type, we prove that the normalizer and the commensurator of AX in AS are equal and are the product of AX by the quasi-centralizer of AX in AS . Looking to the associated monoids A + S and A + X , we describe the quasi-centralizer of A+X in A + S thanks to results in Coxete...

Let X be a non-degenerate left Banach module over a normed algebra A having a bounded approximate left identity. We show that, if A is a left ideal of a larger algebra, then this representation can be extended to a representation of the larger algebra. Based on this result, we study in detail the existence and properties of representations of the various centralizer algebras of A which are comp...

The normalizer NW (WJ) of a standard parabolic subgroup WJ of a finite Coxeter group W splits over the parabolic subgroup with complement NJ consisting of certain minimal length coset representatives of WJ in W . In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type Dn) the centralizer CW (w) of an element w ∈ W is in a simil...