XXXXXXXXXXExercise 4 1 (i) Let Cn = {e, x, ..., xn-1} (x n = e) be the cyclic cyclic group of order n and let H be a subgroup of Cn. By considering min{i > 0 : x i ? H} show that H = hx mi for some m|n. *(ii) Find all subgroups of C18, explaining your answer. 2(i) Find (a XXXXXXXXXXmod 17), (b) 4-1 (mod 17), (c XXXXXXXXXXmod 17) *(ii) Find (a XXXXXXXXXXmod 13), (b XXXXXXXXXXmod 13) 3 Let G be a group in which g 2 = e for all g ? G. Show that G is abelian. *4 Thoughout this question, let G be a finite group and H and K subgroups of G. (i) Prove that H n K is a subgroup of G. (ii) Let HK denote the set {hk : h ? H, k ? K}. Prove that if H n K = {e} then |HK| = |H||K|. (iii) Prove that if G is abelian then HK is a subgroup of G. (iv) Prove that if |G| = 150, |H| = 6, |K| = 25 then G = HK. *5 Let G and H be groups. A map f : G -? H is said to be an isomorphism if f is bijective and f(g1g2) = f(g1)f(g2) for all g1, g2 ? G. If there exists an isomorphism f : G -? H then G and H are said to be isomorphic and we write G ~= H. You can think of isomorphic groups as being the same group with different names: if we write down the group table for G and rename elements we get the group table for H. (i) Show that if f is as above then f(eG) = eH and f(g -1 ) = (f(g))-1 for all g ? G. (Here eG is identity element of G and eH the identity element of H.) (ii) Write down the group tables for C4 and for Z * 5 . Find an isomorphism f : C4 -? Z * 5 . (iii) Recall that if X and Y are sets then the Cartesian product is the set of ordered pairs X × Y = {(x, y) : x ? X, y ? Y }. Show that if G and H are groups then G × H is a group under the group operation (g1, h1) ? (g2, h2) = (g1g2, h1h2). 1 (iv) Write down the group table for C2 × C2. Show that C2 × C2 is not isomorphic to C4. The following questions are optional and harder 6 Let G be a group of order 4. By first considering the order of elements in G, show that either G ~= C4 or G ~= C2 × C2. [Thus, up to isomorphism, there are just two groups of order 4.] 7 Let G be a group of order 6. By first considering the order of elements in G, show that either G ~= C6 or G ~= D3, where D3 is the symmetry group of the triangle, D3 = hx, y : x 3 = e, y2 = e, yx = x 2 yi = {e, x, x2 , y, xy, xy2}. [Thus, up to isomorphism, there are just two groups of order 6.]