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12 - Groups

Published online by Cambridge University Press:  05 September 2012

Alan F. Beardon
Affiliation:
University of Cambridge
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Summary

Groups

In Chapter 1 we defined what it means to say that a set G is a group with respect to an operation *, and we studied groups of permutations of a set. In this chapter we shall study ‘abstract’ groups. Although this may seem like a more difficult task, every ‘abstract’ group is the permutation group of some set so that, in some sense, this apparent change of direction is only an illusion.

We have already seen many examples of groups: the sets ℤ, ℝ and ℂ of integers, real numbers, and complex numbers, respectively, the spaces ℝn and ℂn, the set of matrices of a fixed size, and indeed any vector space, all form a group with respect to addition. Likewise, the set ℝ+ of positive real numbers, the set ℂ* of non-zero complex numbers, the set of complex numbers of modulus one, the set of non-singular n × n matrices, and the set of n-th roots of unity, all form a group with respect to multiplication. Finally, the set of bijections of a given set onto itself, the set of isometries of ℂ, and of ℝn, the permutations of {1, …, n}, and the set of non-singular (invertible) linear transformations of a vector space onto itself, all form a group with respect to the usual composition of functions (that is, if f * g is the function defined by (f * g)(x) = f(g(x))).

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Algebra and Geometry , pp. 215 - 253
Publisher: Cambridge University Press
Print publication year: 2005

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  • Groups
  • Alan F. Beardon, University of Cambridge
  • Book: Algebra and Geometry
  • Online publication: 05 September 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800436.013
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  • Groups
  • Alan F. Beardon, University of Cambridge
  • Book: Algebra and Geometry
  • Online publication: 05 September 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800436.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Groups
  • Alan F. Beardon, University of Cambridge
  • Book: Algebra and Geometry
  • Online publication: 05 September 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800436.013
Available formats
×