Editing Coset (section)

=== Normal subgroups ===
A subgroup {{math|''N''}} of a group {{math|''G''}} is a [[normal subgroup]] of {{math|''G''}} if and only if for all elements {{mvar|g}} of {{math|''G''}} the corresponding left and right cosets are equal, that is, {{math|1=''gN'' = ''Ng''}}. This is the case for the subgroup {{mvar|H}} in the first example above. Furthermore, the cosets of {{math|''N''}} in {{math|''G''}} form a group called the [[quotient group|quotient group or factor group]] {{math|''G''{{hsp}}/{{hsp}}''N''}}.

If {{math|''H''}} is not [[normal subgroup|normal]] in {{math|''G''}}, then its left cosets are different from its right cosets. That is, there is an {{mvar|a}} in {{math|''G''}} such that no element {{mvar|b}} satisfies {{math|1=''aH'' = ''Hb''}}. This means that the partition of {{math|''G''}} into the left cosets of {{math|''H''}} is a different partition than the partition of {{math|''G''}} into right cosets of {{math|''H''}}. This is illustrated by the subgroup {{mvar|T}} in the first example above. (''Some'' cosets may coincide. For example, if {{mvar|a}} is in the [[center (group theory)|center]] of {{math|''G''}}, then {{math|1=''aH'' = ''Ha''}}.)

On the other hand, if the subgroup {{math|''N''}} is normal the set of all cosets forms a group called the quotient group {{math|''G''{{hsp}}/{{hsp}}''N''}} with the operation {{math|∗}} defined by {{math|1=(''aN'') ∗ (''bN'') = ''abN''}}. Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".