\begin{definition}[Definition:Exact Normal Series]
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'''.
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\end{definition}