For a finite group $G$, we write **Type - length** as follows

- If $G$ is abelian then we write "Abelian - $n$", where $n$ is the number of cyclic groups in the unique cyclic decomposition of $G$ as $C_{d_1} \times C_{d_2}\times \cdots \times C_{d_n}$ with $d_i\mid d_{i+1}$ and $d_i>0$.
- If $G$ is nilpotent but not abelian then we write "Nilpotent - $n$", where $n$ is the nilpotency class of $G$.
- If $G$ is solvable but not nilpotent then we write "Solvable - $n$", where $n$ is the derived length of $G$.
- If $G$ is not solvable then we write "Non-solvable - $n$", where $n$ is the number of Jordan factors of $G$.

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**Knowl status:**

- Review status: reviewed
- Last edited by Jennifer Paulhus on 2022-07-19 08:50:59

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**History:**(expand/hide all)

- 2023-06-13 16:28:42 by David Roe
- 2022-07-19 08:50:59 by Jennifer Paulhus (Reviewed)
- 2020-12-06 11:58:49 by Manami Roy
- 2020-12-05 12:50:18 by David Roe
- 2020-12-05 12:49:01 by David Roe
- 2020-12-02 15:52:11 by John Jones
- 2020-12-02 15:51:53 by John Jones
- 2020-12-01 20:07:21 by Manami Roy

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