scholarly journals THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

2021 ◽  
pp. 1-70
Author(s):  
DAVID GEPNER ◽  
JEREMIAH HELLER
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Independent Interest ◽  
Splitting Theorem ◽  
Motivic Homotopy Theory ◽  
Group A ◽  
Motivic Homotopy ◽  
A Finite Group

Abstract We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

scholarly journals On groups admitting a noncyclic abelian automorphism group

1973 ◽  
Vol 9 (3) ◽  
pp. 363-366 ◽  
Author(s):  
J.N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Automorphism Group ◽  
Fixed Points ◽  
Free Operator ◽  
A Finite Group

It is shown that a condition of Kurzwell concerning fixed-points of certain operators on a finite group G is sufficient to ensure that G is soluble. The result generalizes those of Martineau on elementary abelian fixed-point-free operator groups.

scholarly journals Motivic connective K-theories and the cohomology of A(1)

2011 ◽  
Vol 7 (3) ◽  
pp. 619-661 ◽  
Author(s):  
Daniel C. Isaksen ◽  
Armira Shkembi
Keyword(s):  
Spectral Sequence ◽  
Fixed Points ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
The Real ◽  
Special Cases ◽  
Ext Groups ◽  
Motivic Homotopy Theory ◽  
Motivic Homotopy ◽  
Homotopy Fixed Points

AbstractWe make some computations in stable motivic homotopy theory over Spec ℂ, completed at 2. Using homotopy fixed points and the algebraic K-theory spectrum, we construct over ℂ a motivic analogue of the real K-theory spectrum KO. We also establish a theory of motivic connective covers over ℂ to obtain a motivic version of ko. We establish an Adams spectral sequence for computing motivic ko-homology. The E2-term of this spectral sequence involves Ext groups over the subalgebra A(1) of the motivic Steenrod algebra. We make several explicit computations of these E2-terms in interesting special cases.

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.

scholarly journals Finite Groups and Lie Rings with an Automorphism of Order 2n

2016 ◽  
Vol 60 (2) ◽  
pp. 391-412
Author(s):  
E. I. Khukhro ◽  
N. Yu. Makarenko ◽  
P. Shumyatsky
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Finite Groups ◽  
Derived Length ◽  
Lie Rings ◽  
Soluble Subgroup ◽  
A Finite Group

AbstractSuppose that a finite groupGadmits an automorphismof order 2nsuch that the fixed-point subgroupof the involutionis nilpotent of classc. Letm=) be the number of fixed points of. It is proved thatGhas a characteristic soluble subgroup of derived length bounded in terms ofn,cwhose index is bounded in terms ofm,n,c. A similar result is also proved for Lie rings.

A GENERALIZED FIXED-POINT-FREE ACTION

2012 ◽  
Vol 12 (03) ◽  
pp. 1250172
Author(s):  
İSMAİL Ş. GÜLOĞLU ◽  
GÜLİN ERCAN
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Solvable Group ◽  
Upper Bound ◽  
Prime Order ◽  
Free Action ◽  
Group A ◽  
Point Free Action ◽  
A Finite Group ◽  
Coprime Order

In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ CG(A) of prime order or of order 4, every conjugate of x in G is also contained in CG(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.

scholarly journals The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

2021 ◽  
pp. 1-52
Author(s):  
M. ALI ASADI-VASFI ◽  
NASSER GOLESTANI ◽  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Positive Part ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Cuntz Semigroup ◽  
Fixed Point Algebra ◽  
Stably Finite ◽  
A Finite Group

Abstract Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\text{Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha}$ be the fixed point algebra. Then the radius of comparison satisfies ${\text{rc}} (A^{\alpha }) \leq {\text{rc}} (A)$ and ${\text{rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{\text{card} (G))} \cdot {\text{rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\text{Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\text{Cu}} (A)$ , and the purely positive part of ${\text{Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\text{rc}} (A)> 0$ , ${\text{rc}} (A^{\alpha }) = {\text{rc}} (A)$ , and ${\text{rc}} (C^* (G, A, \alpha ) = ( {1}/{2}) {\text{rc}} (A)$ .

scholarly journals Finite Groups with Certain Permutability Criteria

2019 ◽  
Vol 12 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Rola A. Hijazi ◽  
Fatme M. Charaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if itpermutes with all Sylow subgroups of G. In this note we prove that if P, the Sylowp-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.

On Conjugate-Permutable Subgroups of a Finite Group

2005 ◽  
Vol 12 (04) ◽  
pp. 669-676 ◽  
Author(s):  
Mingyao Xu ◽  
Qinhai Zhang
Keyword(s):  
Finite Group ◽  
Necessary Conditions ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called conjugate-permutable in G if HHg = HgH for any g ∈ G. A group G is called an ECP-group if every subgroup of G is conjugate-permutable in G. In this paper, we study the influence of conjugate-permutable subgroups on the structure of a finite group, especially on the nilpotency or supersolvability of the group, and give some sufficient or necessary conditions for a finite group to be an ECP-group.

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

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