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CLASS NUMBERS AND GROUPS OF ALGEBRAIC GROUPS
Mathematics of the USSR-Izvestiya, 1980The class number of an algebraic group G defined over a global field is the number of double cosets of the adele group GA with respect to the subgroups of integral and principal adeles. In most cases the set of double cosets has the natural structure of an abelian group, called the class group of G.
Platonov, V. P. +2 more
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Mathematical Notes, 2002
Let \(B\) be a class of groups. The elementary class with base \(B\) (denote it by \({\mathfrak E}(B)\)) is defined as the minimal class of groups containing \(B\) and closed with respect to the following four standard operations: (S) taking subgroups; (Q) taking quotient groups; (E) group extensions; (L) taking direct (inductive) limits.
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Let \(B\) be a class of groups. The elementary class with base \(B\) (denote it by \({\mathfrak E}(B)\)) is defined as the minimal class of groups containing \(B\) and closed with respect to the following four standard operations: (S) taking subgroups; (Q) taking quotient groups; (E) group extensions; (L) taking direct (inductive) limits.
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Quotient Groups of Groups of Certain Classes
Ukrainian Mathematical Journal, 2000Summary: For an arbitrary variety \(\mathfrak X\) of groups and an arbitrary class \(\mathfrak Y\) of groups that is closed under quotient groups we prove that a quotient group \(G/N\) of the group \(G\) possesses an invariant system with \(\mathfrak X\)- and \(\mathfrak Y\)-factors (respectively, is residually a \(\mathfrak Y\)-group) if \(G ...
Chernikov, N. S., Trebenko, D. Ya.
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Journal of Mathematical Sciences, 2014
The author considers subclasses of the class of all abelian MAP groups und studies under which hypotheses the mapping \(G\to G^\wedge\) from the subclass in the class of abelian groups is injective. \(G^\wedge\) denotes the character group of \(G\). He proves that for the class of all locally quasi--convex groups the mapping is not injective.
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The author considers subclasses of the class of all abelian MAP groups und studies under which hypotheses the mapping \(G\to G^\wedge\) from the subclass in the class of abelian groups is injective. \(G^\wedge\) denotes the character group of \(G\). He proves that for the class of all locally quasi--convex groups the mapping is not injective.
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Minimal classes and maximal class inp-groups
Israel Journal of Mathematics, 1999The authors study finite non-commutative \(p\)-groups \(G\) with the property that the number of minimal classes (i.e., the smallest non-singleton conjugacy classes) is \(p-1\). Among others, they prove the following results. For such a group, \(Z_2(G)\) is an elementary Abelian \(p\)-group, \(|Z_2(G):Z(G)|=p\) and the minimal classes are exactly the ...
LONGOBARDI, Patrizia +2 more
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ON A CLASS OF CYCLICALLY PRESENTED GROUPS
International Journal of Algebra and Computation, 2004This paper considers the problem for what n and ω is the cyclically presented group Gn(ω) irreducible and trivial and studies the case when [Formula: see text] and n≥5.
Martin Edjvet, Paul Hammond
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From Class Groups to Class Fields
1999Since the first printing of the book [35] by H. Zassenhaus and the author in 1989 algorithmic algebraic number theory has attracted rapidly increasing interest. This is documented, for example, by a regular meeting ANTS (algebraic number theory symposium) every two years whose proceedings [1], [6] give a good survey about ongoing research.
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Selmer groups and ideal class groups
Commentarii mathematici Universitatis Sancti Pauli = Rikkyo Daigaku sugaku zasshi, 1993The author proves bounds on the rank of an elliptic curve over an arbitrary number field \(k\) containing the cube roots of unity, in case the curve has a rational point of order 3. The bound involves the degree of \(k\), and the difference between the class group of \(k\) and part of the class group of the field \(K\) obtained by adjoining the ...
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The Ideal Class Group and the Unit Group
1977Recall that the ideal class group of a number ring R consists of equivalence classes of nonzero ideals under the relation $$ I\sim J\quad iff\quad \alpha I = \beta J\quad for{\text{ }}some{\text{ }}non{\text{ }}zero{\text{ }}\alpha ,\beta \in R; $$ the group operation is multiplication defined in the obvious way, and the fact that this is ...
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