Central extensions and Schur’s multiplicators of Galois groups
نویسندگان
چکیده
منابع مشابه
Galois Groups of Maximal ̂ -extensions
Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th roo...
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Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.
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We study a class of algebra extensions which usually appear in the study of restricted Lie algebras or various quantum objects at roots of unity. The present paper was inspired by the theory of nonrestricted representations of restricted Lie algebras and the theory of quantum groups at roots of unity where algebras are usually finitely generated modules over their centers. Our objective is to d...
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Given a group G and an integer n ≥ 2 we construct a new group e K(G, n). Although this construction naturally occurs in the context of finding new invariants for complex algebraic surfaces, it is related to the theory of central extensions and the Schur multiplier. A surprising application is that Abelian groups of odd order possess naturally defined covers that can be computed from a given cov...
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ژورنال
عنوان ژورنال: Nagoya Mathematical Journal
سال: 1983
ISSN: 0027-7630,2152-6842
DOI: 10.1017/s0027763000020389