# Definition:Exact Normal Series

## Definition

Let $\sequence {H_i}_{i \mathop \in I \mathop \subseteq \Z}$ 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 $i \in I$:

$\Img {\phi_i} = \map \ker {\phi_{i + 1} }$

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

Then $\sequence {H_i}$ is referred to as exact at $H_i$.

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

## Also known as

An exact series is also known as an exact sequence.