# Related constructions and facts

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