# Definition:Solvable Group

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## Definition

Let $G$ be a finite group.

Then $G$ is a **solvable group** if and only if it has a composition series in which each factor is a cyclic group.

## Also known as

A **solvable group** is also known as a **soluble group**.

## Examples

### Symmetry Group of Equilateral Triangle is Solvable

The symmetry group $D_3$ of the equilateral triangle is a solvable group.

## Also see

- Results about
**solvable groups**can be found**here**.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 75$. Solvable Groups - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**soluble group**(*US*:**solvable group)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**soluble group**(*US*:**solvable group)**