Multiplicative Subgroups of Finite Index in a Division Ring
نویسندگان
چکیده
منابع مشابه
Counting Finite Index Subgroups
Let F be a finitely generated group. Denote by an(T) (resp.<rn(r)) the number of subgroups of F of index n (resp. of index at most n). This paper deals with the connection between the algebraic structure of the group F and the arithmetic properties of the sequence an(F), n — 1,2,3,..., e.g., the growth of the sequence an(T) ("the subgroup growth") or the properties of the function Cr(s) = X2£Li...
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One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...
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This theorem generalizes the (well-known) fact that the multiplicative group of a finite field is cyclic. Most proofs of this fact can actually be used to prove Theorem 1 in all its generality, so there is not much need to provide another proof here. But yet, let us sketch a proof of Theorem 1 that requires only basic number theory. The downside is that it is very ugly. First, an easy number-th...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1994
ISSN: 0002-9939
DOI: 10.2307/2159872