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Chapter 3 Quotient Groups
Motivation: Normal Subgroups and Quotient Groups. Normal subgroups and quotients are intimately related. As we will see, quotients groups can only be formed using normal subgroups. Normal subgroups are also fundamentally connected to group homomorphisms; kernels are always normal subgroups, and, as it turns out, normal subgroups are always kernels as well!
A group that contains no non-trivial normal subgroups is called simple . The classification of finite simple groups is one of the pioneering achievements in mathematics.
Quotients are essential in group theory for several reasons, primarily related to understanding the structure of groups, simplifying problems, and building new mathematical objects from existing ones.
The Isomorphism Theorems. On of the seminal theorems of group theory that we will prove is the First Isomorphism Theorem. Without getting into too many details, the First Isomorphism Theorem shows that every homomorphism ’induces’ a isomorphism between a quotient group of the domain and a subgroup of the codomain. This allows mathematicians to study the properties of homomorphisms by examining their kernels and images.
Classifying Groups. The presence (or absence) of normal subgroups in a group helps determine its structure. Simple groups, which have no non-trivial normal subgroups, are the building blocks for all finite groups.
Understanding ’How Close’ a Group is to Being Abelian. Abelian groups are some of the most thoroughly understood objects in group theory. Thus, understanding the ways in which more complex groups ’behave’ like abelian groups provides powerful insights into their structures and inner workings. Normal series, poly properties
