New Groups from Old and Groups within Groups

This chapter has two main themes and one subsidiary, all inter-related. Section 1–7 deal with direct product groups, sections 8–12 with quotient groups, and section 13 with equivalence classes. Quotient groups were first mentioned on an experimental basis in section 3.4. Those introductory ideas are again used in the first portion of this chapter. A more detailed and rigorous account of quotient groups follows in the second portion. The reader who prefers not to postpone the rigorous approach could read sections 8–12 before sections 1–7. The final section of the chapter is an elementary treatment of equivalence, for the benefit of those readers who may not have previously met this important concept in its general algebraic sense; it underlies the discussion of modular arithmetic in section 2.1 and is essential to the understanding of the rôle of cosets in quotient groups which is developed in the next chapter.