Editing Coset (section)

=== 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''}}]].