Silvain Rideau 1091 Evans silvain.rideau@berkeley.edu www.normalesup.org/~srideau/en/teaching Homework 4 Due October 2nd The questions indicated as Harder will not be taken in account when grading. Problem 1 : Let G R. 1. Assume that for all b R>0 , there exists g G such that 0 < g < b. Show that for all x, y R such that x < y, there is g G such that x < g < y. 2. (Harder) If a = inf{g G g > 0} 0, show that G = aZ. Problem 2 : 1. Show that x e2ix is group homomorphism from R to C . 2. Let T = {x C x = 1} (here x denotes the absolute value). Show that R/Z T. 3. Show that in Q/Z R/Z all elements have finite order but the order can be arbitrarily large. 4. Show that Q/Z = {x C xn = 1 for some n Z>0 } T. Note that the group was called Z in the previous homework. Problem 3 : Let G be a group and H G such that [G H] = n < 1. Assume that H P G, show that for all g G, g n H. 2. (Harder) Find a counterexample when H is not normal. Problem 4 : Let n Z, n 3 and dn. Let r denote one of the rotations in D2n and H = rd . 1. Show that H P D2n . 2. If d = 1, show that D2n /H Z/2Z. 3. If d = 2 (in particular, n has to be even), D2n /H Z/2Z Z/2Z. 4. If d > 2, D2n /H D2d . 1