scholarly journals An extension theorem for integral representations

1997 ◽  
Vol 63 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Finite Group ◽  
Matrix Representation ◽  
Extension Theorem ◽  
Iwasawa Theory ◽  
Integral Representations ◽  
Roots Of Unity ◽  
Cyclotomic Fields ◽  
Qualitative Result ◽  
Class Groups ◽  
A Finite Group

AbstractBy a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

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).

scholarly journals Permutation characters

1972 ◽  
Vol 13 (1) ◽  
pp. 76-90 ◽  
Author(s):  
Fiona M. Ross
Keyword(s):  
Finite Group ◽  
Matrix Representation ◽  
Complex Field ◽  
Permutation Matrices ◽  
A Finite Group ◽  
Permutation Character

We suppose throughout that G is a finite group with a faithful matrix representation X over the complex field. We suppose that X affords a character π of degree r whose values are rational (hence rational integers). If the matrices in some representation of G affording a character π0 are all permutation matrices, then π0 is called a permutation character. Permutation characters have non-negative integral values. In the general case, we consider what properties of permutation characters are true of π, and in particular, under what circumstances π is a permutation character. Note that assuming X to b faithful is equivalent to considering the image group X(G) instead of G.

On the Character Rings of Finite Groups

1963 ◽  
Vol 15 ◽  
pp. 605-612 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Roots Of Unity ◽  
Closed Field ◽  
One Dimensional ◽  
Elementary Group ◽  
A Finite Group ◽  
Character Ring

The characters of the representations of a finite group G over a field K of characteristic zero generate a ring oK(G) of functions on G, the K-character ring of G, which is readily seen to be Zϕ1 + . . . + Zϕn, where Z is the ring of rational integers and ϕ1, . . . , ϕn are the characters of the different irreducible representations of G over K. The theorem that every irreducible representation of G over an algebraically closed field Ω of characteristic zero is equivalent to a representation of G over the subfield of Ω which is generated by the g0th roots of unity (g0 the exponent of G) was proved by Brauer (4) via the theorems that(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations (3).

Integral Representations of the Direct Product of Groups

1963 ◽  
Vol 15 ◽  
pp. 625-630 ◽  
Author(s):  
Alfredo Jones
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Principal Ideal ◽  
Integral Representations ◽  
Principal Ideal Domain ◽  
Linear Combinations ◽  
Quotient Field ◽  
A Finite Group ◽  
Torsion Free ◽  
Product Of Groups

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.

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.

scholarly journals LANDAU’S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS

2016 ◽  
Vol 104 (1) ◽  
pp. 37-43
Author(s):  
MARK L. LEWIS
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Character Theory ◽  
Finite Solvable Group ◽  
Roots Of Unity ◽  
Prime Power Order ◽  
A Finite Group

When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

The Dual of Frobenius' Reciprocity Theorem

1973 ◽  
Vol 25 (5) ◽  
pp. 1051-1059 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Reciprocity Theorem ◽  
Integral Representations ◽  
Irreducible Representations ◽  
Frobenius Reciprocity ◽  
Multiplication Tables ◽  
Symmetry Properties ◽  
Structure Constants ◽  
A Finite Group

In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.

scholarly journals A Generalization of a Theorem of Swan with Applications to Iwasawa Theory

2018 ◽  
Vol 72 (3) ◽  
pp. 656-675
Author(s):  
Andreas Nickel
Keyword(s):  
Finite Group ◽  
Iwasawa Theory ◽  
Finitely Generated ◽  
Global Fields ◽  
A Finite Group

AbstractLet $p$ be a prime and let $G$ be a finite group. By a celebrated theorem of Swan, two finitely generated projective $\mathbb{Z}_{p}[G]$-modules $P$ and $P^{\prime }$ are isomorphic if and only if $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$ and $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$ are isomorphic as $\mathbb{Q}_{p}[G]$-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.

Restricted Determinantal Homomorphisms and Locally Free Class Groups

1990 ◽  
Vol 42 (4) ◽  
pp. 646-658 ◽  
Author(s):  
Victor Snaith
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Number Field ◽  
Class Group ◽  
Class Groups ◽  
A Finite Group

Let K be a number field and let OK denote the integers of K. The locally free class groups, Cl(OK[G]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants, which map the NGH-invariant idèlic units of Ok([Hab] to Cl(OK[G]). These homomorphisms are constructed by means of the Horn-description of Cl(OK[G]), which describes the locally free class group in terms of the representation theory of G, and the technique of Explicit Brauer Induction, which was introduced in [5].

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