# Definition:Normal Series/Factor Group

## Definition

Let $G$ be a group whose identity is $e$.

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$

The factor groups of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$:

$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

are the quotient groups:

$G_1 / G_0, G_2 / G_1, \ldots, G_i / G_{i - 1}, \ldots, G_n / G_{n-1}$

## Also known as

A factor group is also referred to as a factor.