[b]Group[/b]: A [i]group[/i] is a set with a binary operation (*; e.g., composition) that satisfies the following four properties:[br][list=1][*][b]Closure[/b]: A set is [i]closed[/i] under a binary operation if, when any two elements of the set are combined under that binary operation, the result is another element of the set.[/*][*][b]Identity[/b]: The [i]identity[/i] element in a set is an element, e, of that set such that, for all elements a in the set, e*a=a*e=a.[/*][*][b]Inverses[/b]: For any element a in the set, there exists an [i]inverse[/i] element, a[sup]-1[/sup], such that a*a[sup]-1 [/sup]= a[sup]-1 [/sup]*a=[sup] [/sup]e.[/*][*][b]Associativity[/b]: If a, b, and c are elements of the set, then (a*b)*c=a*(b*c).[br][/*][/list]

Prove that the set of isometries of the plane, under the operation of composition, forms a group. Can you identify any subgroups (subsets of the set of isometries that also form a group under composition)? You can use the applet below to investigate the inverses and compositions of various isometries.