2: Groups
- Page ID
- 85708
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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 \(G\times G \to G\) 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 \(\ast \colon S\times S\to S\) to denote a binary operation on a set \(S\) that sends the pair \((x,y)\) to \(x\ast y\text{.}\) Recall that a binary operation \(\ast\) is associative means that \(x\ast(y\ast z)= (x\ast y)\ast z\) for all \(x,y,z\in S\text{.}\)