The Schur Subgroup of a p-Adic Field

1979 ◽  
Vol 31 (2) ◽  
pp. 300-303
Author(s):  
Eugene Spiegel ◽  
Allan Trojan
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Group Algebras ◽  
Brauer Group ◽  
Ramification Index ◽  
Simple Component ◽  
Image Position ◽  
Cyclotomic Algebras ◽  
A Finite Group ◽  
Tame Ramification

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only ifwhere c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

1988 ◽  
Vol 108 (1-2) ◽  
pp. 117-132
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Simple Group ◽  
Finite Groups ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Nilpotency Index ◽  
Characteristic 3 ◽  
A Finite Group

SynopsisLet J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

scholarly journals Twisted group algebras and their representations

1964 ◽  
Vol 4 (2) ◽  
pp. 152-173 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Group Algebras ◽  
Linear Combinations ◽  
Twisted Group Algebra ◽  
Formula One ◽  
Twisted Group ◽  
Image Position ◽  
A Finite Group ◽  
Group Product

Let be a finite group, a field. A twisted group algebra A() on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of (AB), where AB is the group product of A, B ∈: . One gets the ordinary group algebra () by taking each fA, B ≠ 1.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

scholarly journals On the determination of the ramification index in Clifford's theorem

1988 ◽  
Vol 31 (3) ◽  
pp. 469-474
Author(s):  
Robert W. van der Waall
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Ramification Index ◽  
A Finite Group

Let K be a field, G a finite group, V a (right) KG-module. If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.Suppose N is a normal subgroup of G. The KN-module VN is not irreducible in general, even when V is irreducible as KG-module. A part of the well-known theorem of A. H. Clifford [1, V.17.3] yields the following.

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

scholarly journals A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals ON THE MORITA REDUCED VERSIONS OF SKEW GROUP ALGEBRAS OF PATH ALGEBRAS

2020 ◽  
Vol 71 (3) ◽  
pp. 1009-1047
Author(s):  
Patrick Le Meur
Keyword(s):  
Finite Group ◽  
Monoidal Category ◽  
Path Algebra ◽  
Group Algebras ◽  
Explicit Formulas ◽  
Morita Equivalent ◽  
Path Algebras ◽  
A Finite Group ◽  
Skew Group Algebras ◽  
Skew Group Algebra

Abstract Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.

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