Groups

  • John Stillwell
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The concept of group came up briefly in Section 6.4, when we observed the following three properties of one-to-one functions from a set to itself:
$$\begin{gathered} {g_1}({g_2}{g_3}) = ({g_1}{g_2}){g_3}{\text{ (associativity)}} \hfill \\ {\text{g1 = 1g (identity)}} \hfill \\ {\text{g}}{{\text{g}}^{ - 1}} = {g^{ - 1}}g = 1(inverse) \hfill \\ \end{gathered} $$
where 1 denotes the identity function and g−1 denotes the inverse of function g. This is perhaps an unusual way to introduce the group concept, but it does help explain the prevalence of groups in mathematics. Sets and functions are the raw material for the construction of all abstract mathematical objects and, among the major concepts of algebra, the group concept is closest to pure set theory. This is confirmed by Cayley’s theorem (Section 7.2), which shows that any binary relation with the associative, identity and inverse properties can be modelled by composition of one-to-one functions.

Keywords

Abelian Group Normal Subgroup Automorphism Group Permutation Group Dihedral Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • John Stillwell
    • 1
  1. 1.Department of MathematicsMonash UniversityClaytonAustralia

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