Another important class of groups is the simple groups, whose only normal subgroups are \({G}\) and \({\mathbf{1}}\). The finite simple groups behave in many ways like primes, and after a long effort have been fully classified. Some additional constructions and facts concerning quotients include:

• The index of a group \({G}\) over a subgroup \({H}\), denoted \({|G:H|}\), is the number of cosets of \({H}\) in \({G}\)
• \({|G:H|=|G|/|H|}\) for finite groups
• \({G/Z(G)\cong\textrm{Inn}(G)}\), the group of all inner automorphisms of \({G}\)
• \({G/Z(G)}\) is cyclic, \({G}\) is abelian
• The abelian simple groups are \({\mathbb{Z}_{p}}\) for \({p}\) prime
• Feit-Thompson theorem: non-abelian simple groups have even order