Algebra: Abstract and Modern

4

Subgroups and Quotient Groups

4.1 Subgroups

4.2 Cyclic Groups

4.3 Cosets of a Subgroup

4.4 Lagrange’s Theorem

4.5 Normal Subgroups

4.6 Quotient Groups

A nonempty set together with certain operations on it is called an algebraic system. In the study of any algebraic system, the subsets of the underlying set which are closed under the operations of the algebraic system called subsystems or substructures. In general, we are not interested in arbitrary subsets of the underlying set A in the algebraic system; for, they do not reflect the fact that A has an algebraic structure imposed on it. Whatever subsets we do consider will be those endowed with algebraic properties derived from those of the given algebraic system. In this chapter, we introduce ...