Titlebook: An Introduction to the Theory of Groups; Joseph J. Rotman Textbook 1995Latest edition Springer Science+Business Media New York 1995 Abelia

arthrodesis

https://doi.org/10.1007/978-3-531-91657-6The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.

Lignans

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Ingratiate

Symmetric Groups and ,-Sets,The definition of group arose from fundamental properties of the symmetric group S.. But there is another important feature of S.: its elements arc functions acting on some underlying set, and this aspect is not explicit in our presentation so far. The notion of . is the appropriate abstraction of this idea.

GIBE

The Sylow Theorems,The order of a group . has consequences for its structure. A rough rule of thumb is that the more complicated the prime factorization of |.|, the more complicated the group. In particular, the fewer the number of distinct prime factors in |G|, the more tractible it is. We now study the “local” case when only one prime divides |.|.

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Normal Series,We begin this chapter with a brief history of the study of roots of polynomials. Mathematicians of the Middle Ages, and probably those in Babylonia, knew the . giving the roots of a quadratic polynomial .(.) = .. + . + .. Setting . transforms .(.) into a polynomial g(.) with no . term:

极深

Extensions and Cohomology,A group . having a normal subgroup . can be “factored” into . and .. The study of extensions involves the inverse question: Given . ⊲ . and ., to what extent can one recapture .?

GNAW

Abelian Groups,Commutativity is a strong hypothesis, so strong that all finite abelian groups are completely classified. In this chapter, we focus on finitely generated and, more generally, countable abelian groups.

胶水

Free Groups and Free Products,The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.

减弱不好

The Word Problem,Novikov, Boone, and Britton proved, independently, that there is a finitely presented group ℬ for which no computer can ever exist that can decide whether an arbitrary word on the generators of ℬ is 1. We shall prove this remarkable result in this chapter.