> [!definition] > > Let $\bracs{G_i}_1^{n}$ be a sequence of [[Group|groups]] and $f_i: G_i \to G_{i + 1}$ be [[Group Homomorphism|group homomorphisms]]. Then the sequence > $ > G_1 \overset{f_1}{\to} G_2 \to \cdots \overset{f_{n - 1}}{\to}G_n > $ > is **exact** if $\im {f_i} = \ker (f_{i + 1})$ for any $1 \le i \le n - 2$. > [!example] > > If the sequence > $ > \bracs{e} \to G' \overset{f}\to G \overset{g}\to G'' \to 0 > $ > is **exact**, then $f$ is injective and $g$ is surjective. Moreover, if $H = \ker g$, then there exists a commutative diagram > $ > \begin{CD} > \bracs{e} @>>> G' @>{f}>> G @>{g}>> G'' @>>> \bracs{e}\\ > @. @VVV @VVV @VVV \\ > \bracs{e} @>>> H @>>> G @>>> G/H @>>> \bracs{e} > \end{CD} > $ > where the columns are exact, and rows are isomorphisms.