Modular and p-adic integral representations of a direct product of groups

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.

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UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

Journal of Symbolic Logic ◽

10.1017/jsl.2014.54 ◽

2014 ◽

Vol 79 (4) ◽

pp. 1001-1019 ◽

Cited By ~ 2

Author(s):

ASHER M. KACH ◽

ANTONIO MONTALBÁN

Keyword(s):

Direct Product ◽

Disjoint Union ◽

Second Order ◽

Linear Orders ◽

Second Order Arithmetic ◽

Order Arithmetic ◽

Natural Classes ◽

Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

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On the Schur multiplicator of a central quotient of a direct product of groups

Selecta ◽

10.1007/978-3-642-61708-9_47 ◽

1987 ◽

pp. 623-632

Author(s):

P. J. Hilton ◽

U. Stammbach

Keyword(s):

Direct Product ◽

Central Quotient ◽

Product Of Groups

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On Nilpotent Products of Cyclic Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-039-x ◽

1960 ◽

Vol 12 ◽

pp. 447-462 ◽

Cited By ~ 15

Author(s):

Ruth Rebekka Struik

Keyword(s):

Abelian Group ◽

Direct Product ◽

Finite Abelian Group ◽

Nilpotent Groups ◽

Multiplication Table ◽

Direct Products ◽

Free Products ◽

Cyclic Groups ◽

Special Cases ◽

Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

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Certain Groups of Orthonormal Step Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1957-049-x ◽

1957 ◽

Vol 9 ◽

pp. 413-425 ◽

Cited By ~ 26

Author(s):

J. J. Price

Keyword(s):

Abelian Group ◽

Direct Product ◽

Compact Abelian Group ◽

Walsh Functions ◽

Cyclic Groups ◽

Step Functions ◽

Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

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