2.7: Summaries of Groups We've Seen
When you see the following groups in the wild, you should assume they are equipped with the following default operations, unless otherwise noted. You should know:
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What elements the groups contain;
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What their default operations are;
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Their orders (and, if they're infinite, whether they're countably infinite or uncountable);
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Whether or not they are abelian.
| Group(s) | Operation |
|---|---|
| \(\mathbb{Z}\text{,}\) \(\mathbb{Q}\) | Addition of Numbers |
| \(n\mathbb{Z}\) | Addition of Numbers |
| \(\mathbb{R}\text{,}\) \(\mathbb{C}\) | Addition of Numbers |
| \(\mathbb{Q}^*\text{,}\) \(\mathbb{Q}^+\) | Multiplication of Numbers |
| \(\mathbb{R}^*\text{,}\) \(\mathbb{R}^+\text{,}\) \(\mathbb{C}^*\) | Multiplication of Numbers |
| \(\mathbb{M}_{m\times n}(\mathbb{R}), \mathbb{M}_n(\mathbb{R})\) | Matrix Addition |
| \(GL(n,\mathbb{R}), SL(n,\mathbb{R})\) | Matrix Multiplication |
| \(\mathbb{Z}_n\) | Addition mod \(n\) |
| \(\mathbb{Z}_2^2\) | Componentwise Addition mod 2 |
| \(F\) (see Example \(2.6.6\)) | Pointwise Addition |
| \(B\) (see Example \(2.6.7\)) | Composition |