# Central series

This article defines a property that can be evaluated for a subgroup series

View a complete list of properties of subgroup series

## Definition

### Central series of finite length

This is the default meaning of the term central series.

$G = K_1 \ge K_2 \ge \dots K_c \ge K_{c+1} = 1$

is termed a central series if it satisfies the following conditions:

1. It is a normal series: every $K_i$ is normal in $G$
2. For every $i$, $K_i/K_{i+1}$ is contained in the center of $G/K_{i+1}$.

Equivalently, it should satisfy the condition that for every $i$:

$[G,K_i] \subset K_{i+1}$