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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