152 CHAPTER 4 Some Mathematical Topics on Which to Practice Proof Techniques Isomorphism and Subgroups One of the goals mentioned earlier was to investigate when groups of the same size have the same structure and when they do not. Let's revisit it. It was easy to conclude that all groups of size one have the same structure, because the only element present must be the identity element. All groups of size two have the same structure, and they are abelian. Let's explicitly compare 1 2 1 2 and f = . The ðP 2 , Þ = S 2 , ð 2 , + 2 Þ, and ð à , × 3 Þ: The set P 2 has two elements, t = 3 1 2 2 1 set 2 has two elements, [0] 2 and [1] 2 (the index 2 refers to the fact that these are congruence classes (mod 2)). The set à = f½1 3 , ½2 3 g also has two elements (the index 3 refers to the fact that these are con- 3 gruence classes (mod 3)). Let's consider their operation tables: j f j f f f + 2 ½0 2 j ½1 2 j ½0 2 ½1 2 ½0 2 ½1 2 ½1 2 ½1 2 × 3 ½1 3 ½2 3 ½1 3 j ½1 3 ½2 3 ½2 3 j ½2 3 ½1 3 Table for 3 Table for S 2 Table for 2 We can see that these groups' tables are identical, except for the different names of the elements. They are said to be isomorphic (this word is made of two Greek words, iso, "the same" and morphos, "shape"). From the point of view of abstract algebra, isomorphic groups have the same properties and they are indistinguishable.