# Definition:Exact Normal Series

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## 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**.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Problem $\text{FF}$