scholarly journals Pseudo-reflection group actions on local rings

1982 ◽  
Vol 88 ◽  
pp. 161-180 ◽  
Author(s):  
Luchezar L. Avramov
Keyword(s):  
Finite Group ◽  
Polynomial Ring ◽  
Representation Theorem ◽  
Regular Representation ◽  
Reflection Group ◽  
The Other ◽  
Local Rings ◽  
Natural Representation ◽  
Classical Paper ◽  
A Finite Group

In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk(S1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X1,…,Xn] over a field k of characteristic zero. He proved that S is free as a graded SH-module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.

scholarly journals On the realization of the Gelfand character of a finite group as a twisted trace

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jorge Soto-Andrade ◽  
Maria-Francisca Yáñez-Valdés
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Regular Representation ◽  
Natural Representation ◽  
Fixed Point Set ◽  
Linear Automorphism ◽  
Point Set ◽  
A Finite Group ◽  
The Right ◽  
Complex Characters

Abstract We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

scholarly journals On the power graph of a finite group

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1201-1208 ◽  
Author(s):  
M. Mirzargar ◽  
A.R. Ashrafi ◽  
M.J. Nadjafi-Arani
Keyword(s):  
Finite Group ◽  
The Other ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P(G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

scholarly journals Finite automorphism groups of laminated near-rings

1983 ◽  
Vol 26 (3) ◽  
pp. 297-306 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Complex Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
The Other ◽  
Complex Polynomial ◽  
Infinite Groups ◽  
Nonsingular Matrices ◽  
A Finite Group

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

Invariants of hyperplane groups and vanishing ideals of finite sets of points

2012 ◽  
Vol 55 (2) ◽  
pp. 355-367 ◽  
Author(s):  
H. E. A. Campbell ◽  
Jianjun Chuai
Keyword(s):  
Finite Group ◽  
Polynomial Ring ◽  
Constructive Proof ◽  
Additive Subgroup ◽  
Finite Sets ◽  
Vanishing Ideal ◽  
Invariant Ring ◽  
Invariant Differential ◽  
A Finite Group ◽  
Vanishing Ideals

AbstractWe define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ $W\subseteq\mathbb{F}^n$, the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

scholarly journals On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three

2021 ◽  
Vol 50 (2) ◽  
pp. 549-557
Author(s):  
Suzila Mohd Kasim ◽  
Athirah Nawawi
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Symplectic Group ◽  
Undirected Graph ◽  
The Other ◽  
Symplectic Groups ◽  
Main Work ◽  
Commuting Graph ◽  
Mathematical Formulas ◽  
A Finite Group

Suppose G be a finite group and X be a subset of G. The commuting graph, denoted by C(G,X), is a simple undirected graph, where X ⊂G being the set of vertex and two distinct vertices x,y∈X are joined by an edge if and only if xy = yx. The aim of this paper was to describe the structure of disconnected commuting graph by considering a symplectic group and a conjugacy class of elements of order three. The main work was to discover the disc structure and the diameter of the subgraph as well as the suborbits of symplectic groups S4(2)', S4(3) and S6(2). Additionally, two mathematical formulas are derived and proved, one gives the number of subgraphs based on the size of each subgraph and the size of the conjugacy class, whilst the other one gives the size of disc relying on the number and size of suborbits in each disc.

RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250157 ◽  
Author(s):  
B. TOLUE ◽  
A. ERFANIAN
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
The Other ◽  
Commuting Graph ◽  
Other Hand ◽  
A Finite Group

The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

scholarly journals Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5323-5334 ◽  
Author(s):  
Asma Hamzeh ◽  
Ali Ashrafi
Keyword(s):  
Finite Group ◽  
Characteristic Polynomial ◽  
Finite Groups ◽  
The Other ◽  
Laplacian Spectrum ◽  
Simple Graphs ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

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