ABSTRACT
In this chapter we introduce the reader to a group structure. In Section 2.1, we present basic definitions, examples and terminology related to groups. In Section 2.2, we define a subgroup, give many classic examples and present a shortcut for verifying subgroup. In Section 2.3, we present an important class of groups called cyclic groups. Permutation groups are an essential part of group theory and are presented in Section 2.4. In Section 2.5, we create new groups and subgroups by means of a product. In Section 2.6, we introduce functions between groups called homomorphisms. As a follow up, in Section 2.7, we use isomorphisms to define what it means for groups to be essentially equal (or isomorphic). In Section 2.8, we define perhaps one of the most equivalence relations on a group whose equivalence classes are called cosets, and these cosets can sometimes form a group. In Section 2.9, we investigate exactly when cosets form a group and look at several important examples. In Section 2.10, we investigate further notmal subgroups and consider groups which have no normal subgroups. Finally, in Section 2.11, we prove some fundamental isomorphism results for groups and factor groups.
