scholarly journals Galois Theory with Infinitely Many Idempotents

1969 ◽  
Vol 35 ◽  
pp. 83-98 ◽  
Author(s):  
O.E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Linear Independence ◽  
Commutative Rings ◽  
Classical Field ◽  
Fixed Field ◽  
Noncommutative Rings ◽  
Real Change ◽  
A Finite Group

In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

scholarly journals Galois Theory for Rings with Finitely Many Idempotents

1966 ◽  
Vol 27 (2) ◽  
pp. 721-731 ◽  
Author(s):  
O. E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Ring Extensions ◽  
A Finite Group

In [5], Chase, Harrison and Rosenberg proved the Fundamental Theorem of Galois Theory for commutative ring extensions S ⊃ R under two hypotheses: (i) 5 (and hence R) has no idempotents except 0 and l; and (ii) 5 is Galois over R with respect to a finite group G—which in the presence of (i) is equivalent to (ii′): S is separable as an R-algebra, finitely generated and projective as an R-module, and the fixed ring under the group of all R-algebra automorphisms of S is exactly R.

scholarly journals Actions of finite groups on commutative rings I

2003 ◽  
Vol 34 (3) ◽  
pp. 201-212
Author(s):  
A. G. Naoum ◽  
W. K. Al-Aubaidy
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Finite Groups ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
The Ideal

Let $R$ be a commutative ring with 1, and let $G$ be a finite group of automorphisms of $R$. Denote by $R^G$ the fixed subring of $G$, and let $I$ be a subset of $R^G$. In this paper we prove that if the ideal generated by $I$ in $R$ satisfies a certain property with regard to projectivity, flatness, multiplication or related concepts, then the ideal generated by $I$ in $R^G$ also satisfies the same property.

Real class sizes and real character degrees

2010 ◽  
Vol 150 (1) ◽  
pp. 47-71 ◽  
Author(s):  
ROBERT M. GURALNICK ◽  
GABRIEL NAVARRO ◽  
PHAM HUU TIEP
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Character Degrees ◽  
Real Character ◽  
Fixed Field ◽  
Field Of Values ◽  
A Finite Group

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

ON THE IDENTITIES OF REPRESENTATIONS OF FINITE GROUPS OVER COMMUTATIVE RINGS

1992 ◽  
Vol 02 (01) ◽  
pp. 103-116
Author(s):  
SAMUEL M. VOVSI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Finite Basis ◽  
Finitely Based ◽  
Commutative Noetherian Ring ◽  
Representations Of Finite Groups ◽  
A Finite Group ◽  
Finite Exponent

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS

2011 ◽  
Vol 10 (05) ◽  
pp. 835-847 ◽  
Author(s):  
ANTONIO PAQUES ◽  
VIRGÍNIA RODRIGUES ◽  
ALVERI SANT'ANA
Keyword(s):  
Finite Group ◽  
Galois Theory ◽  
Group Actions ◽  
Galois Extensions ◽  
Partial Action ◽  
Partial Group ◽  
Azumaya Algebra ◽  
Unital Ring ◽  
One To One ◽  
A Finite Group

Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

scholarly journals A Note on Counting Flows in Signed Graphs

10.37236/7958 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Matt DeVos ◽  
Edita Rollová ◽  
Robert Šámal
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Fundamental Theorem ◽  
Signed Graph ◽  
Signed Graphs ◽  
Odd Order ◽  
A Finite Group ◽  
Special Case

Tutte initiated  the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $\Gamma$, let $\epsilon_2(\Gamma)$ be the largest integer $d$ so that $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $\Gamma$-flows in $G$ for every abelian group $\Gamma$ with $\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Beck and Zaslavsky [JCTB 2006] had previously established the special case of this result when $d=0$ (i.e., when $\Gamma$ has odd order).

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.

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