The structure of a partial Galois extension II

2016 ◽  
Vol 15 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Finite Group ◽  
Galois Extension ◽  
Structure Theorem ◽  
Galois Extensions ◽  
Partial Action ◽  
A Finite Group ◽  
Partial Galois Extension

Let [Formula: see text] be a partial Galois extension where [Formula: see text] is a partial action of a finite group on a ring [Formula: see text] such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in [Formula: see text], and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for [Formula: see text].

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 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 Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

scholarly journals On central commutator Galois extensions of rings

2000 ◽  
Vol 24 (5) ◽  
pp. 289-294
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Structure Theorem ◽  
Projective Group ◽  
Galois Extensions ◽  
Separable Extensions

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

Very Exceptional Group

2014 ◽  
Vol 1006-1007 ◽  
pp. 1071-1075
Author(s):  
Xiao Yu Liang ◽  
Xin Zhang
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Nilpotent Group ◽  
Galois Extension ◽  
Prime Order ◽  
Zeta Functions ◽  
Number Fields ◽  
Galois Groups ◽  
Exceptional Group ◽  
A Finite Group

<p>A finite group is called exceptional if for a Galois extension of number fields with the Galois groups , the zeta function of between and does not appear in the Brauer-Kuroda relation of the Dedekind zeta functions. Furthermore, a finite group is called very exceptional if its nontrivial subgroups are all exceptional. In this paper，a Nilpotent group is very exceptional if and only if it has a unique subgroup of prime order for each divisor of .</p>

Galois corings and groupoids acting partially on algebras

2020 ◽  
pp. 2140003
Author(s):  
S. Caenepeel ◽  
T. Fieremans
Keyword(s):  
Galois Extension ◽  
Galois Theory ◽  
Type Theorem ◽  
Partial Action ◽  
Morita Theory ◽  
Maschke Type Theorem ◽  
Galois Corings ◽  
Partial Galois Extension ◽  
Special Case

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

On three special types of partial Galois extensions

2019 ◽  
Vol 19 (12) ◽  
pp. 2150004
Author(s):  
Xiaolong Jiang ◽  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Galois Extension ◽  
Galois Extensions ◽  
Partial Galois Extension

In this paper, we study the following three special types of partial Galois extensions: DeMeyer–Kanzaki partial Galois extension, partial Galois Azumaya extension and commutator partial Galois extension.

scholarly journals Skew group rings which are Galois

2000 ◽  
Vol 23 (4) ◽  
pp. 279-283
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Extension ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Automorphism ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

LetS*Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS*Gis anG′-Galois extension of(S*G)G′, whereG′is the inner automorphism group ofS*Ginduced by the elements inG, thenSis aG-Galois extension ofSG. A necessary and sufficient condition is also given for the commutator subring of(S*G)G′inS*Gto be a Galois extension, where(S*G)G′is the subring of the elements fixed under each element inG′.

SEMIALGEBRAS AND THEIR ALGEBRAS OF DIFFERENCES WITH PARTIAL GROUP ACTIONS ON THEM

2013 ◽  
Vol 06 (03) ◽  
pp. 1350038
Author(s):  
Ram Parkash Sharma ◽  
Anu
Keyword(s):  
Finite Group ◽  
Prime Ideal ◽  
Group Actions ◽  
Rocky Mountain ◽  
Prime Ideals ◽  
Central Idempotent ◽  
Group Rings ◽  
Partial Action ◽  
Partial Group ◽  
A Finite Group

Let A be a semialgebra defined in [R. P. Sharma, Anu and N. Singh, Partial group actions on semialgebras, Asian European J. Math.5(4) (2012), Article ID:1250060, 20pp.] over an additively cancellative and commutative semiring K. In additively cancellative semirings, the subtractive ideals play an important role. If P is a subtractive and G-prime ideal of an additively cancellative and yoked semiring A, where G is a finite group acting on A, then A has finitely many n(≤ |G|) minimal primes over P (see [R. P. Sharma and T. R. Sharma, G-prime ideals in semirings and their skew group rings, Comm. Algebra34 (2006) 4459–4465], Lemma 3.6, a result analogous to Lemma 3.2 of Passman [D. S. Passman, It's essentially Maschke's theorem, Rocky Mountain J. Math.13 (1983) 37–54]). Consider a subtractive partial action α of a finite group G on A such that each Dg is generated by a central idempotent 1g of A and the intersection D = ⋂g∈G,Dg≠0Dg of nonzero Dg's is nonzero. It is not necessary that number of minimal primes in Spec A over a subtractive and α-prime ideal P of a yoked semiring A is less than or equal to the order of the group, if 1d ∈ P (Example 3.2). However, we show that the result is true if 1d ∉ P (Corollary 3.1). We also study the prime ideals of the partial fixed subsemiring Aα of A.

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