Definition:Exact Normal Series

Definition
Let $\left \langle {H_i}\right \rangle_{i \in I \subseteq \Z}$ be a normal series:
 * $\cdots \stackrel {\phi_{i-1}}{\longrightarrow} H_{i-1} \stackrel {\phi_i}{\longrightarrow} H_i \stackrel {\phi_{i+1}}{\longrightarrow} H_{i+1} \stackrel {\phi_{i+2}}{\longrightarrow} \cdots$

Suppose that, for some $i \in I$:
 * $\operatorname{Im} \left({\phi_i}\right) = \ker \left({\phi_{i+1}}\right)$

That is, the image of one homomorphism is the kernel of the next.

Then $\left \langle {H_i}\right \rangle$ is referred to as exact at $H_i$.

If $\left \langle {H_i}\right \rangle$ is exact for all $i \in I$, then $\left \langle {H_i}\right \rangle$ itself is an exact normal series.

Also known as
An exact series is also known as an exact sequence.