Solvable Groups

# Solvable Groups

Definition: Let $G$ be a group. Then $G$ is said to be a Solvable group if there exists a (finite) chain of successive subgroups $\{ 1 \} = G_0 \leq G_1 \leq ... \leq G_n = G$ with the following properties:1) $G_i$ is a normal subgroup of $G_{i+1}$ for all $0 \leq i \leq n - 1$.2) $G_{i+1}/G_i$ is an abelian group for all $0 \leq i \leq n - 1$. |

The following proposition tells us that solvable groups are plentiful in that every abelian group is solvable.

Proposition 1: Let $G$ be an abelian group. Then $G$ is a solvable group. |

**Proof:**Suppose that $G$ is an abelian group. Then $\{ 1 \} = G_0 \leq G_1 = G$ is a chain of successive subgroups. Note that $G_0 = \{ e \}$ is a normal subgroup of $G_1 = G$ trivially. Furthermore, $G_1/G_0 = G/\{ e \} \cong G$ is an abelian group. Thus $G$ is a solvable group. $\blacksquare$