Definition:Exact Sequence of Groups

Definition
Let $(G,\cdot)$ be a group.

Consider the sequence of groups $G_i$ and group homomorphisms $\varphi_i$:
 * $\displaystyle \cdots \stackrel{\varphi_{i-2}}{\longrightarrow} G_{i-1} \stackrel{\varphi_{i-1}}{\longrightarrow} G_i \stackrel{\varphi_i}{\longrightarrow} G_{i+1} \stackrel{\varphi_{i+1}}{\longrightarrow} \cdots$

The sequence is said to be exact if $\mathrm{im} \left( \varphi_i \right) = \mathrm{ker} \left( \varphi_{i+1} \right)$ for all $i$.