Editing Coset (section)

== More examples ==

=== Integers ===
Let {{math|''G''}} be the [[additive group]] of the integers, {{math|1='''Z''' = ({..., −2, −1, 0, 1, 2, ...}, +)}} and {{math|''H''}} the subgroup {{math|1=(3'''Z''', +) = ({..., −6, −3, 0, 3, 6, ...}, +)}}. Then the cosets of {{math|''H''}} in {{math|''G''}} are the three sets {{math|3'''Z'''}}, {{math|3'''Z''' + 1}}, and {{math|3'''Z''' + 2}}, where {{math|1=3'''Z''' + ''a'' = {{mset|..., −6 + ''a'', −3 + ''a'', ''a'', 3 + ''a'', 6 + ''a'', ...}}}}. These three sets partition the set {{math|'''Z'''}}, so there are no other right cosets of {{mvar|H}}. Due to the [[commutivity]] of addition {{math|1=''H'' + 1 = 1 + ''H''}} and {{math|1=''H'' + 2 = 2 + ''H''}}. That is, every left coset of {{mvar|H}} is also a right coset, so {{mvar|H}} is a normal subgroup.<ref>{{harvnb|Fraleigh|1994|loc=p. 117}}</ref> (The same argument shows that every subgroup of an Abelian group is normal.<ref name=Fraleigh>{{harvnb|Fraleigh|1994|loc=p. 169}}</ref>)

This example may be generalized. Again let {{math|''G''}} be the additive group of the integers, {{math|1='''Z''' = ({..., −2, −1, 0, 1, 2, ...}, +)}}, and now let {{math|''H''}} the subgroup {{math|1=(''m'''''Z''', +) = ({..., −2''m'', −''m'', 0, ''m'', 2''m'', ...}, +)}}, where {{mvar|m}} is a positive integer. Then the cosets of {{math|''H''}} in {{math|''G''}} are the {{mvar|m}} sets {{math|''m'''''Z'''}}, {{math|''m'''''Z''' + 1}}, ..., {{math|''m'''''Z''' + (''m'' − 1)}}, where {{math|1=''m'''''Z''' + ''a'' = {{mset|..., −2''m'' + ''a'', −''m'' + ''a'', ''a'', ''m'' + ''a'', 2''m'' + ''a'', ...}}}}. There are no more than {{mvar|m}} cosets, because {{math|1=''m'''''Z''' + ''m'' = ''m''('''Z''' + 1) = ''m'''''Z'''}}. The coset {{math|(''m'''''Z''' + ''a'', +)}} is the [[Modular arithmetic#Congruence classes|congruence class]] of {{mvar|a}} modulo {{mvar|m}}.<ref>{{harvnb|Joshi|1989|loc= p. 323}}</ref> The subgroup {{math|''m'''''Z'''}} is normal in {{math|'''Z'''}}, and so, can be used to form the quotient group {{math|'''Z'''{{hsp}}/{{hsp}}''m'''''Z'''}} the group of [[Integers mod n|integers mod {{math|''m''}}]].

=== Vectors ===
Another example of a coset comes from the theory of [[vector space]]s. The elements (vectors) of a vector space form an [[abelian group]] under [[vector addition]]. The [[linear subspace|subspaces]] of the vector space are [[subgroups]] of this group. For a vector space {{math|''V''}}, a subspace {{math|''W''}}, and a fixed vector {{math|'''a'''}} in {{math|''V''}}, the sets
<math display="block">\{\mathbf{x} \in V \mid \mathbf{x} = \mathbf{a} + \mathbf{w}, \mathbf{w} \in W\}</math>
are called [[affine subspace]]s, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional [[Euclidean space|geometric]] vectors, these affine subspaces are all the "lines" or "planes" [[Parallel (geometry)|parallel]] to the subspace, which is a line or plane going through the origin. For example, consider the [[Plane (geometry)|plane]] {{math|'''R'''<sup>2</sup>}}. If {{mvar|m}} is a line through the origin {{mvar|O}}, then {{mvar|m}} is a subgroup of the abelian group {{math|'''R'''<sup>2</sup>}}. If {{mvar|P}} is in {{math|'''R'''<sup>2</sup>}}, then the coset {{math|''P'' + ''m''}} is a line {{math|''m''′}} parallel to {{mvar|m}} and passing through {{mvar|P}}.<ref>{{harvnb|Rotman|2006|loc=p. 155}}</ref>

=== Matrices ===
Let {{mvar|G}} be the multiplicative group of matrices,<ref>{{harvnb|Burton|1988|loc=pp. 128, 135}}</ref>
<math display="block">G = \left \{\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix} \colon a, b \in \R, a \neq 0 \right\},</math>
and the subgroup {{mvar|H}} of {{mvar|G}},
<math display="block">H= \left \{\begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \colon c \in \mathbb{R} \right\}.</math>
For a fixed element of {{mvar|G}} consider the left coset
<math display="block">\begin{align}
\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix} H = &~ \left \{\begin{bmatrix} a & 0 \\ b & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \colon c \in \R \right\} \\
=&~ \left \{\begin{bmatrix} a & 0 \\ b + c & 1 \end{bmatrix} \colon c \in \mathbb{R}\right\} \\
=&~ \left \{\begin{bmatrix} a & 0 \\ d & 1 \end{bmatrix} \colon d \in \mathbb{R}\right\}.
\end{align}</math>
That is, the left cosets consist of all the matrices in {{mvar|G}} having the same upper-left entry. This subgroup {{mvar|H}} is normal in {{mvar|G}}, but the subgroup
<math display="block">T= \left \{\begin{bmatrix} a & 0 \\ 0 & 1 \end{bmatrix} \colon a \in \mathbb{R} - \{0\} \right\}</math>
is not normal in {{mvar|G}}.