Isomorphism of noncommutative group algebras of torsion-free groups over a field
نویسنده
چکیده
The isomorphism problem for group algebras over a field with arbitrary characteristic of some special classes of torsion-free non-abelian groups is explored. Specifically, the following are proved: Suppose F is a field and G is a torsion-free group with centre C(G) such that FG ∼= FH as F -algebras for any group H . Then it is shown that H is torsion-free (provided that FG is without zero divisors), and if G is soluble we have even more that C(G) ∼= C(H). The latter extends classical results due to Higman(1940)-May(1969) when G is torsion-free abelian. Moreover, if G is an R-group or a D-group, then the above F isomorphism yields that so is H . Subject Classification: 20C05, 16S35, 16U60, 16W20, 20E, 20F.
منابع مشابه
2 3 N ov 2 00 6 The problem of the classification of the nilpotent class 2 torsion free groups up
In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more c...
متن کاملA brief introduction to quaternion matrices and linear algebra and on bounded groups of quaternion matrices
The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...
متن کاملISOMORPHISM OF MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS WITH SEMI-COMPLETE p-PRIMARY COMPONENTS
Let G be a p-mixed abelian group with semi-complete torsion subgroup Gt such that G is splitting or is of torsion-free rank one, and let R be a commutative unitary ring of prime characteristic p. It is proved that the group algebras RG and RH are R-isomorphic for any group H if and only if G and H are isomorphic. This isomorphism relationship extends our earlier results in (Southeast Asian Bull...
متن کاملThe Algebra of Discrete Torsion
We analyze the algebraic structures of G–Frobenius algebras which are the algebras associated to global group quotient objects. Here G is any finite group. These algebras turn out to be modules over the Drinfeld double of the group ring k[G]. We furthermore prove that discrete torsion is a universal group action of H(G, k∗) on G–Frobenius algebras by isomorphisms of the underlying linear struct...
متن کاملNoncommutative Clusters
In this paper we introduce noncommutative clusters and their mutations, which can be viewed as vast generalizations of both “classical” and quantum cluster structures. Each noncommutative cluster X is built on a torsion-free group G and a certain collection of its automorphisms. We assign to X a noncommutative algebra A(X) related to the group algebra of G, which is an analogue of the cluster a...
متن کامل