Silvain Rideau 1091 Evans silvain.rideau@berkeley.edu www.normalesup.org/~srideau/eng/teaching Homework 4 Due February 25th Problem 1 : Let G be a group, p a prime dividing G and X = {(x0 , . . . , xp1 ) Gp i xi = 1}. Let E be the relation dened on X by (x0 , . . . , xp1 )E(y0 , . . . , yp1 ) if there exists k Z/pZ such that yi+k = xi (the indexes are considered to live in Z/pZ, so if i + k p, what we mean is yi+kp = xi ). 1. Show that X = Gp1 . 2. Show that E is an equivalence relation. 3. Show that an equivalence class of E contains a single element if and only if this element is of the form (x, . . . , x) where xp = 1. 4. Show that the equivalence classes of E that are not singletons have cardinal p. 5. Show that p divides the number of equivalence classes of E that are singletons. 6. Show that there exists an element x G of order p. Problem 2 : Let G be a group of order 6. By the previous problem, there is an element a G of order 2 and an element b G of order 3. 1. Show that G = {ai bj i = 0, 1 and j = 0, 1, 2}. 2. Show that aba is either b or b2 . 3. Show that, if aba = b, then G Z/6Z. 4. Show that, if aba = b2 , then G D6 . 1