2: Groups
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- 2.1: Examples of groups
- Groups are one of the most basic algebraic objects, yet have structure rich enough to be widely useful in all branches of mathematics and its applications. A group is a set G with a binary operation G×G→G that has a short list of specific properties. Before we give the complete definition of a group in the next section, this section introduces examples of some important and useful groups.
- 2.2: Definition of a group
- We will use the notation ∗:S×S→S to denote a binary operation on a set S that sends the pair (x,y) to x∗y. Recall that a binary operation ∗ is associative means that x∗(y∗z)=(x∗y)∗z for all x,y,z∈S.