# Subnormal series

of a group $G$

A subgroup series of $G$, $$E = G_0 \le G_1 \le \cdots \le G_n = G$$ where each subgroup $G_i$ is a normal subgroup of $G_{i+1}$. The quotient groups $G_{i+1}/G_i$ are called factors, and the number $n$ is called the length of the subnormal series. Infinite subnormal series have also been studied (see Subgroup system). A subnormal series that cannot be refined further is called a composition series, and its factors are called composition factors.