2.7: Summaries of Groups We've Seen
- Page ID
- 84858
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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 |