Editing Quotient group (section)

=== Complex integer roots of 1 ===
[[File:Normal subgroup illustration.svg|right|thumb|The cosets of the fourth [[roots of unity]] ''N'' in the twelfth roots of unity ''G''.]]
The twelfth [[roots of unity]], which are points on the [[Complex number|complex]] [[unit circle]], form a multiplicative abelian group {{tmath|1= G }}, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup <math>N</math> made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group <math>G\,/\,N</math> is the group of three colors, which turns out to be the cyclic group with three elements.