Editing Exact sequence (section)

=== Integers modulo two ===
Consider the following sequence of abelian groups:
:<math>\mathbf{Z} \mathrel{\overset{2\times}{\,\hookrightarrow}} \mathbf{Z} \twoheadrightarrow \mathbf{Z}/2\mathbf{Z}</math>

The first homomorphism maps each element ''i'' in the set of integers '''Z''' to the element 2''i'' in '''Z'''. The second homomorphism maps each element ''i'' in '''Z''' to an element ''j'' in the quotient group; that is, {{nowrap|''j'' {{=}} ''i'' mod 2}}. Here the hook arrow <math>\hookrightarrow</math> indicates that the map 2× from '''Z''' to '''Z''' is a monomorphism, and the two-headed arrow <math>\twoheadrightarrow</math> indicates an epimorphism (the map mod 2).  This is an exact sequence because the image 2'''Z''' of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as 

:<math>2\mathbf{Z} \mathrel{\,\hookrightarrow} \mathbf{Z} \twoheadrightarrow \mathbf{Z}/2\mathbf{Z}</math>

In this case the monomorphism is  2''n'' ↦ 2''n'' and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to 2'''Z'''. The image of 2'''Z''' through this monomorphism is however exactly the same subset of '''Z''' as the image of '''Z''' through ''n'' ↦ 2''n'' used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2'''Z''' is not the same set as '''Z''' even though the two are isomorphic as groups.

The first sequence may also be written without using special symbols for monomorphism and epimorphism:
:<math>0 \to \mathbf{Z} \mathrel{\overset{2\times}{\longrightarrow}} \mathbf{Z} \longrightarrow \mathbf{Z}/2\mathbf{Z} \to 0</math>

Here 0 denotes the trivial group, the map from '''Z''' to '''Z''' is multiplication by 2, and the map from '''Z''' to the [[factor group]] '''Z'''/2'''Z''' is given by reducing integers [[modular arithmetic|modulo]] 2. This is indeed an exact sequence:
* the image of the map 0 → '''Z''' is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first '''Z'''.
* the image of multiplication by 2 is 2'''Z''', and the kernel of reducing modulo 2 is also 2'''Z''', so the sequence is exact at the second '''Z'''.
* the image of reducing modulo 2 is '''Z'''/2'''Z''', and the kernel of the zero map is also '''Z'''/2'''Z''', so the sequence is exact at the position '''Z'''/2'''Z'''.

The first and third sequences are somewhat of a special case owing to the infinite nature of '''Z'''. It is not possible for a [[finite group]] to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the [[first isomorphism theorem]] is 

:<math>1 \to N \to G \to G/N \to 1</math>
(here the trivial group is denoted <math>1,</math> as these groups are not supposed to be [[abelian group|abelian]]).

As a more concrete example of an exact sequence on finite groups:

:<math>1 \to C_n \to D_{2n} \to C_2 \to 1</math>

where <math>C_n</math> is the [[cyclic group]] of order ''n'' and <math>D_{2n}</math> is the [[dihedral group]] of order 2''n'', which is a non-abelian group.