J-Equivalence of group representations

1971 ◽  
Vol 70 (1) ◽  
pp. 9-14 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Finite Group ◽  
Galois Group ◽  
Complex Representation ◽  
Group Representations ◽  
Root Of Unity ◽  
Representation Ring ◽  
A Finite Group

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).

scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

scholarly journals Mapping class group representations from Drinfeld doubles of finite groups

2020 ◽  
Vol 29 (05) ◽  
pp. 2050033
Author(s):  
Jens Fjelstad ◽  
Jürgen Fuchs
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Group Representations ◽  
Torelli Group ◽  
Group Data ◽  
A Finite Group ◽  
Marked Points

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group [Formula: see text], focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have finite images, and that for surfaces of genus at least [Formula: see text] their restriction to the Torelli group is non-trivial if and only if [Formula: see text] is non-abelian.

Finite Linear Groups of Degree Seven. I

1969 ◽  
Vol 21 ◽  
pp. 1042-1053 ◽  
Author(s):  
David B. Wales
Keyword(s):  
Finite Group ◽  
Complex Representation ◽  
Linear Groups ◽  
Subsequent Paper ◽  
Prime Degree ◽  
Sylow Group ◽  
A Finite Group ◽  
Finite Linear

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

scholarly journals On the kernels of representations of finite groups

1981 ◽  
Vol 22 (2) ◽  
pp. 151-154 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Modular Representation ◽  
Complex Representation ◽  
Representations Of Finite Groups ◽  
A Finite Group

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:(1) G is p-solvable of p-length 1.(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

Finite Linear Groups of Prime Degree

1969 ◽  
Vol 21 ◽  
pp. 1025-1041 ◽  
Author(s):  
David B. Wales
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Irreducible Representations ◽  
Complex Representation ◽  
Linear Groups ◽  
Prime Degree ◽  
Long Time ◽  
A Finite Group ◽  
Finite Linear

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Classification of Finite Group-Frames and Super-Frames

2007 ◽  
Vol 50 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Deguang Han
Keyword(s):  
Finite Group ◽  
Equivalence Classes ◽  
Group Representations ◽  
Exact Number ◽  
A Finite Group ◽  
Frame Representations

AbstractGiven a finite group G, we examine the classification of all frame representations of G and the classification of all G-frames, i.e., frames induced by group representations of G. We show that the exact number of equivalence classes of G-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number L such that there exists an L-tuple of strongly disjoint G-frames.

scholarly journals On tensor factorisation for representations of finite groups

2004 ◽  
Vol 69 (1) ◽  
pp. 161-171 ◽  
Author(s):  
Emanuele Pacifici
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Finite Groups ◽  
Group Structure ◽  
Complex Representation ◽  
Normal Subgroups ◽  
Representations Of Finite Groups ◽  
Projective Representations ◽  
A Finite Group

We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G.

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