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## 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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