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

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