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Discrete Mathematics and Combinatorics by T. Sengadir

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Chapter 8

Groups, Rings and Fields

Learning Objectives

On completing this chapter, you should be able to:

  • recall the definitions of the algebraic structures semigroup, monoid and group

  • give examples of semigroups, monoids and groups

  • prove elementary properties of groups

  • recall the definition of a subgroup

  • identify subgroups of standard groups

  • compute the cosets of subgroups in simple examples

  • state and prove the Lagrange Theorem

  • define a normal subgroup

  • give examples of normal subgroups

  • define the notion of quotient group and give examples

  • state the definitions of homomorphism and isomorphism between groups

  • identify isomorphic groups

  • show that every group G is isomorphic to and subgroup of the group of one-to-one and onto functions on G

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