4.6: Classification Groups
- Page ID
- 132677
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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.
|
Group order |
Abelian |
Non-abelian |
|
|
1 |
{e} |
||
|
2 |
\(C_2=D_1 \) |
||
|
3 |
\(C_3 \) |
||
|
4 |
\(C_4 \) or \(C_2 \times C_2=D_2 \) |
||
|
5 |
\(C_5 \) |
||
|
6 |
\(C_6 \) |
\(S_3=D_3 \) |
|
|
7 |
\(C_7 \) |
||
|
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 \(C_3 \times C_3 \) |
||
|
10 |
\(C_{10} \) |
\(D_5 \) |
|
|
11 |
|||
|
12 |
|||
|
13 |
|||
|
14 |
|||
|
15 |
|||
|
16 |
|||
|
17 |