Exact Sequence of Groups

A sequence of homomorphisms where image equals kernel at each stage

Exact Sequence of Groups

\cdots \longrightarrow G_{i-1}\xrightarrow{f_{i-1}} G_i \xrightarrow{f_i} G_{i+1} \longrightarrow \cdots

is exact at $G_i$ if

\operatorname{im}(f_{i-1}) = \ker(f_i),

where $\operatorname{im}(f_{i-1})$ is the image and $\ker(f_i)$ is the kernel . The sequence is exact if it is exact at every term.

A short exact sequence is an exact sequence of the form

1 \longrightarrow N \xrightarrow{\iota} G \xrightarrow{\pi} Q \longrightarrow 1,

which encodes that $\iota$ is injective, $\pi$ is surjective, and $N\cong \ker(\pi)$ is a normal subgroup of $G$ , with $Q\cong$ the quotient group $G/N$ .

Exact sequences package common situations in a “coordinate-free” way; for example, the first isomorphism theorem can be read as saying every homomorphism fits into a natural short exact sequence.

Examples:

For any homomorphism $\varphi:G\to H$ , the sequence $1\to \ker(\varphi)\to G\to \operatorname{im}(\varphi)\to 1$ is short exact.
The quotient map $G\to G/N$ fits into $1\to N\to G\to G/N\to 1$ when $N$ is normal.
The sequence $\mathbb{Z}\xrightarrow{\times n}\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}\to 0$ is exact (interpreting $0$ as the trivial group in additive notation).