This page contains descriptions of a number of groups that can be used as tests for the truth or otherwise of general group-theoretic statements. (Or rather, it will do when someone writes it.)

A few examples of groups that should certainly be included in any list: cyclic groups, dihedral groups, , , , , linear groups of various kinds and over different fields, the quaternion group, the Heisenberg group, the free group on generators, the free Abelian group on generators, the alternating and symmetric groups, the symmetry groups of Platonic solids. There are undoubtedly more: what an article should do is give brief descriptions of each one (or class) and explain what kinds of properties they have. These explanations should be written for the benefit of a reader who is testing some conjecture about groups.