# Definition:Exact Sequence of Groups

## Definition

Let $\left({G, \circ}\right)$ be a group.

Consider the sequence of groups $\left\langle{G_i}\right\rangle$ and group homomorphisms $\phi_i$:

$\displaystyle \cdots \stackrel{\phi_{i-2}}{\longrightarrow} G_{i-1} \stackrel{\phi_{i-1}}{\longrightarrow} G_i \stackrel{\phi_i}{\longrightarrow} G_{i+1} \stackrel{\phi_{i+1}}{\longrightarrow} \cdots$

$\left\langle{G_i}\right\rangle$ is exact if and only if:

$\forall i: \operatorname{Im} \left({\phi_i}\right) = \ker \left({\phi_{i+1} }\right)$

where:

$\operatorname{Im} \left({\phi_i}\right)$ denotes the image of $\phi_i$
$\ker \left({\phi_{i+1} }\right)$ denotes the kernel of $\phi_{i+1}$.