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Absolute Presentation Of A Group
In mathematics, an absolute presentation is one method of defining a group.B. Neumann, ''The isomorphism problem for algebraically closed groups,'' in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562. Recall that to define a group G by means of a presentation, one specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators. In symbols: :G \simeq \langle S \mid R \rangle. Informally G is the group generated by the set S such that r = 1 for all r \in R. But here there is a tacit assumption that G is the "freest" such group as clearly the relations are satisfied in any homomorphic image of G. One way of being able to eliminate this tacit assumption is by specifying that certain words in S should not be equal to 1. That is we specify a set I, called the set of irrelations, such that i \ne 1 for all i \in I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
