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 $\phi_i$:

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

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

$\forall i: \map {\operatorname{Im}} {\phi_i} = \map \ker {\phi_{i+1}}$

where:

$\map {\operatorname{Im}} {\phi_i}$ denotes the image of $\phi_i$

$\map \ker {\phi_{i+1}}$ denotes the kernel of $\phi_{i+1}$.

Also see

• Definition:Exact Sequence