Abstract
This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of
and a normal subgroup H of G, we prove that
induces a unital partial action
of G/H on the subalgebra of invariants
of S such that
is an
-partial Galois extension of
Second, assuming that G is abelian, we construct a commutative inverse semigroup
whose elements are equivalence classes of
-partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between
and T(G, A), where ρ is a congruence on
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
reduces to the case where G is cyclic. The set of idempotents of
is also investigated.
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