Smith equivalence of representations

1983 ◽  
Vol 94 (1) ◽  
pp. 61-99 ◽  
Author(s):  
Ted Petrie
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Homotopy Sphere ◽  
Smooth Action ◽  
Fixed Set ◽  
A Finite Group

An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set ΣG consisting of two points p and q, are the tangential representations Tp Σ and Tq Σ of G at p and q equal? Put another way: Describe the representations (V, W) of G which occur as (Tp ΣTq Σ) for Σ a sphere with smooth action of G and ΣG = p ∪ q. Under these conditions we say V and W are Smith equivalent (21) and write V ~ W. A stronger equivalence relation is also interesting. We say representations V and W are s-Smith equivalent if (V, W) = (Tp Σ, Tq Σ) and Σ is a semi-linear G sphere (23), i.e. ΣK is a homotopy sphere for all K and ΣG = p ∪ q. In this case we write V ≈ W.

scholarly journals Integral representations with trivial first cohomology groups

1982 ◽  
Vol 85 ◽  
pp. 231-240 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Integral Representations ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Abelian Semigroup ◽  
A Finite Group

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

Locally linear actions on 3-manifolds

1988 ◽  
Vol 104 (2) ◽  
pp. 253-260 ◽  
Author(s):  
SŁawomir Kwasik ◽  
Kyung Bai Lee
Keyword(s):  
Finite Group ◽  
Topological Group ◽  
Group Action ◽  
Smooth Manifold ◽  
Local Behaviour ◽  
Smooth Action ◽  
A Finite Group ◽  
Locally Linear

Let a finite group G act topologically on a closed smooth manifold Mn. One of the most natural and basic questions is whether such an action can be smoothed. More precisely, let γ:G × Mn → Mn be a topological action of G on Mn. The action γ can be smoothed if there exists a smooth action and an equivariant homeomorphism It is well known that for n ≤ 2 every finite topological group action on Mn is smoothable. However already for n = 3 there are examples of topological actions on 3-manifolds which cannot be smoothed (see [1, 2] and references there). All these actions fail to be smoothable because of bad local behaviour.

Systems of equations and generalized characters in groups

1970 ◽  
Vol 22 (5) ◽  
pp. 1040-1046 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Free Group ◽  
Equivalence Relation ◽  
Elementary Proof ◽  
Systems Of Equations ◽  
Recent Application ◽  
Arbitrary Group ◽  
Frobenius Theorem ◽  
A Finite Group

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

scholarly journals On a classification of the function fields of algebraic tori

1975 ◽  
Vol 56 ◽  
pp. 85-104 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Galois Extension ◽  
Function Fields ◽  
Equivalence Classes ◽  
Finitely Generated ◽  
Abelian Semigroup ◽  
Algebraic Tori ◽  
A Finite Group

Let II be a finite group and denote by MII the class of all (finitely generated Z-free) II-modules. In the previous paper [3] we defined an equivalence relation in MII and constructed the abelian semigroup T(II) by giving an addition to the set of all equivalence classes in MII. The investigation of the semigroup T(II) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group II.

scholarly journals Commuting rational functions revisited

2019 ◽  
Vol 41 (1) ◽  
pp. 295-320 ◽  
Author(s):  
FEDOR PAKOVICH
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Rational Function ◽  
Equivalence Relation ◽  
Rational Functions ◽  
A Finite Group

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

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