4.6: Classification Groups
- Page ID
- 132677
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Classification of finite groups
The following table gives the number of distinct groups of finite order. Here \(C_n \) stands for a cyclic group of order \(n \). Cyclic groups are in red.
\(S_4 \)
|
Group order |
Abelian |
Non-abelian |
|
|
1 |
{e} |
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|
2 |
\(C_2=D_1 \) |
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|
3 |
\(C_3 \) |
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|
4 |
\(C_4 \) or \(C_2 \times C_2=D_2 \) |
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|
5 |
\(C_5 \) |
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|
6 |
\(C_6 \) |
\(S_3=D_3 \) |
|
|
7 |
\(C_7 \) |
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|
8 |
\(C_8 \) or \(C_2 \times C_4 \) or \(C_2 \times C_2 \times C_2 \) |
\(D_4 \) or \(Q_8 \) |
|
|
9 |
\(C_9 \) or \(C_3 \times C_3 \) |
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|
10 |
\(C_{10} \) |
\(D_5 \) |
|
|
11 |
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|
12 |
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13 |
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14 |
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15 |
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16 |
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|
17 |