Although there are many different contexts in which they can arise there are only two possible groups with four elements, the Klein four-group and C4.

Reason: Each of the 3 elements apart from the identity has to have an inverse. Either they are all self inverse or just one is self inverse. There cannot be two self inverse elements or none because in these cases it is impossible to complete the group table.

With three elements there is only one possible structure

Reason: The two elements which are not the identity have to be inverses of each other. A little investigation shows that there is no other way to complete the group table.


(a) {1, 2, 3, 4} with multiplication modulo 5 is the group C4.

(b) {1, 5, 7, 11} with multiplication modulo 12 is the Klein 4-group.

(c) {0, 1, 2, 3} with addition modulo 4 is the group C4.