40 Groups 29. If and y are isomorphisms from the cyclic group group and d(a) = y(a), prove that y. 30. Suppose that : Z50Z50 is an automorphism with Determine a formula for d(x). 31. Prove property 1 of Theorem 6.3. 32. Prove property 4 of Theorem 6.3. 33. Referring to Theorem 6.1, prove that T is indeed a p the set G. 34. Prove or disprove that U(20) and U(24) are isomorp 35. Show that the mapping o (a + bi) = a - bi is an aut the group of complex numbers under addition. Sho serves complex multiplication as well-that is, (x for all x and y in C. (This exercise is referred to in C 36. Let G = {a+ bV2 l a, b are rational} and 2b a, b are rational a Н- а Show that G and H K 134 Groups ((a)) = (ak). Because is one-to-one, b = a*. This proves (b) that (a) = G. When the group operation is addition, property 2 of Theorem 6.2 is np(a); property 4 says that an isomorphism between two Ф(па) cyclic groups takes a generator to a generator. Property 6 is quite useful for showing that two groups are not iso morphic. Often b is picked to be the identity. For example, consider C and R*. Because the equation x = 1 has four solutions in C* but only two in R*, no matter how one attempts to define an isomorphism from C* to R*, property 6 cannot hold. Theorem 6.3 Properties of Isomorphisms Acting on Groups Suppose that o is an isomorphism from a group G onto a group G. Then 1. 1 is an isomorphism from G onto G. 2. G is Abelian if and only if G is Abelian. 3. G is cyclic if and only if G is cyclic. 4. If K is a subgroup of G, then (K) = {¢(k) | k E K} is a subgroup of G. 5. If K is a subgroup of G, then 1 (K) {g EGI(g) e K} is a subgroup of G. 6. d(Z(G)) Z(G). PROOF Properties 1 and 4 are left as exercises (Exercises 31 and 32). Properties 2 and 6 are a direct consequence of property 3 of Theorem 6.2. Property 3 follows from property 4 of Theorem 6.2 and property 1 of Theorem 6.3. Property 5 follows from properties 1 and 4. Theorems 6.2 and 6.3 show that isomorphic groups have many prop- erties in common. Actually, the definition is precisely formulated so that isomorphic groups have all group theoretic properties in common. By this we mean that if two groups are isomornhic then any property 1h