Show that there are only two distinct groups with four elements,as follows. Call the elements of the group e, a. b,c.
Let a denote a nonidentity element whose square is the identity.The row and column labeled by e are known. Show that the rowlabeled by a is determined by the requirement that each groupelement must appear exactly once in each row and column; similarly,the column labeled by a is determined. There are now four tableentries left to determine. Show that there are exactly two possibleways to complete the multiplication table that are consistent withthe constraints on multiplication tables. Show that these two waysof completing the table yield the multiplication tables of the twogroups with four elements that we have already encountered.
