Universal finite group extensions and a non-splitting theorem

1973 ◽  
Vol 15 (4) ◽  
pp. 375-383
Author(s):  
John S. Rose
Keyword(s):  
Finite Group ◽  
Splitting Theorem ◽  
Group Extensions

Finite group extensions of irrational rotations

1975 ◽  
Vol 21 (2-3) ◽  
pp. 240-259 ◽  
Author(s):  
William A. Veech
Keyword(s):  
Finite Group ◽  
Irrational Rotations ◽  
Group Extensions

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

On equivalence of finite group extensions

1971 ◽  
Vol 123 (3) ◽  
pp. 191-200 ◽  
Author(s):  
Charles Edward Johnson ◽  
Hans Zassenhaus
Keyword(s):  
Finite Group ◽  
Group Extensions

An amalgamation theorem for finite group extensions

1991 ◽  
Vol 57 (4) ◽  
pp. 325-331 ◽  
Author(s):  
Frieder Haug ◽  
Ulrich Meierfrankenfeld ◽  
Richard E. Phillips
Keyword(s):  
Finite Group ◽  
Group Extensions

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 a Splitting Theorem of Gaschütz

1966 ◽  
Vol 15 (1) ◽  
pp. 57-60 ◽  
Author(s):  
John S. Rose
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Prime Number ◽  
Splitting Theorem

Let G be any finite group, and p any prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote by Kp(G) the unique smallest normal subgroup of G for which the quotient G/Kp(G) is a p-group. G/Kp(G) is called the p-residual of G. W. Gaschütz (2, Satz 7) has proved the followingTheorem. Set K = Kp(G). If the Sylow p-subgroups of K are abelian, then G splits over K.

scholarly journals Finite group extensions and the Baum–Connes conjecture

2007 ◽  
Vol 11 (3) ◽  
pp. 1767-1775 ◽  
Author(s):  
Thomas Schick
Keyword(s):  
Finite Group ◽  
Group Extensions

scholarly journals 1-Cohomology and Splitting of Group Extensions

1976 ◽  
Vol 19 (3) ◽  
pp. 369-371 ◽  
Author(s):  
G. N. Pandya ◽  
R. D. Bercov
Keyword(s):  
Normal Subgroup ◽  
Splitting Theorem ◽  
Abelian Normal Subgroup ◽  
Group Extensions ◽  
Related Theorem

The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))bf(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=nef(e)n-1 for all e ∈ E.

scholarly journals Finite group extensions and the Atiyah conjecture

2007 ◽  
Vol 20 (04) ◽  
pp. 1003-1052 ◽  
Author(s):  
Peter Linnell ◽  
Thomas Schick
Keyword(s):  
Finite Group ◽  
Group Extensions ◽  
Atiyah Conjecture

scholarly journals Finite group extensions of shifts of finite type: -theory, Parry and Livšic

2016 ◽  
Vol 37 (4) ◽  
pp. 1026-1059 ◽  
Author(s):  
MIKE BOYLE ◽  
SCOTT SCHMIEDING
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Finite Type ◽  
Finite Abelian Group ◽  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Group Extensions ◽  
Dynamical Zeta Function ◽  
Periodic Data

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group$G$, Parry showed how to define a$G$-extension$S_{A}$from a square matrix over$\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of$A$over$\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function$\det (I-tA)^{-1}$(which captures the ‘periodic data’ of the extension) would classify the extensions by$G$of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic$\text{K}$-theory group$\text{NK}_{1}(\mathbb{Z}G)$is non-trivial (e.g. for$G=\mathbb{Z}/n$with$n$not square-free) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of$\text{NK}_{1}(\mathbb{Z}G)$, for every non-trivial abelian$G$we show that there exists a shift of finite type with an infinite family of mixing non-conjugate$G$extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for$G$(not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the ‘positive$K$-theory’ setting for positive equivalence of matrices over$\mathbb{Z}G[t]$.

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