1. The commutative inverse semigroup of partial abelian extensions.
- Author
-
Bagio, Dirceu, Cañas, Andrés, Marín, Víctor, Paques, Antonio, and Pinedo, Héctor
- Subjects
COMMUTATIVE algebra ,GALOIS theory ,GROUP algebras ,GROUP theory ,ISOMORPHISM (Mathematics) ,COMMUTATIVE rings ,SUBGROUP growth ,FINITE groups - Abstract
This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action α
G of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF