MA136 Introduction to Abstract Algebra

Homework Assignment 3 Please Enjoy Responsibly! Attempt all the questions on this sheet, and hand in solutions to A2, A4, A5, B1, B2, B3, B4. Solutions to your supervisor's pigeon-loft by 3pm Monday, week 9. Your supervisor will mark some, but not all, of these questions. They don't know which ones yet, so there's no point asking them. (A1) Show that cyclic groups are abelian. (A2) In each of the following groups G, write down the cyclic subgroup generated by g. (a) G = S, g = exp(2i=7). (b) G = Z=12Z, g = 8. (c) G = GL2(R), g = ( 0 1 ??1 0 ). (d) G = R=Z, g = 5=7. (e) G = D4, g = 3. (A3) Which of the following groups G are cyclic? Justify your answer for each, and if G is cyclic then write down a generator. (a) G = kZ (where k is a non-zero integer). (b) G = Z=mZ (where m is a positive integer). (c) D3. (A4) For the following groups G and subgroups H, write down the (left) cosets of H in G and determine the index [G : H]. (a) G = 2Z and H = 6Z. (b) G = U4 and H = U2. (c) G = D4 and H = h1i. (d) G = R and H = fa 2 R : a > 0g. (A5) There are four elements of order 5 in R=Z; nd them. There are eight elements of order 3 in R2=Z2; nd them. (A6) In Z2 we let i = (1; 0) and j = (0; 1) as usual. Write 2Z2 = f(2a; 2b) : a; b 2 Zg. Convince yourself that 2Z2 is a subgroup of Z2 of index 4 and that Z2=2Z2 = f0; i; j; i + jg: Write down an addition table for Z2=2Z2. (B1) In this exercise, you will show using contradiction that R is not cyclic. Suppose that it is cyclic and let g 2 R be a generator. Then R = hgi. In particular, jgj1=2 2 R and so jgj1=2 = gm for some integer m. Show that the only solutions to this equation are g = 1. Where's the contradiction? (B2) Let 2 C and write = rei where r, 2 R and r > 0. Show that S = rS. What does the coset rS represent geometrically? (B3) Let G be a group of order p, where p is a prime number. Let H be a subgroup. Show that H must either equal G or the trivial subgroup f1g (Hint: Use Lagrange's Theorem). Deduce that if g 2 G is not the identity element, then G = hgi. (B4) Let 2 [0; 1). Show that 2 R=Z has nite order if and only if is rational. (C1) Show that every non-zero element of R=Q has innite order. (C2) Let v be a column vector in R2 and H a subgroup of GL2(R). We dene the orbit of v under H to be the set Orb(H; v) = fAv : A 2 Hg: Observe Orb(H; 0) = f0g. Suppose v 6= 0. (i) Let H = fI2 : 2 Rg. Show that H is a subgroup of GL2(R). Describe and sketch Orb(H; v). (ii) Let H = SO2(R) (this is the subgroup of rotation matrices|see Sections IV.7 and IX.4 of the lecture notes). Describe and sketch Orb(H; v). (iii) Finally, let H = GL2(R). With the help of (i) and (ii) explain why Orb(H; v) = R2nf0g. (C3) Let v be a column vector in R2. In the previous assignment we dened the stabilizer of v to be Stab(v) = fA 2 GL2(R) : Av = vg; and showed that it is a subgroup of GL2(R). Now let v and w be non-zero column vectors in R2. (i) Show that there is some B 2 GL2(R) such that Bv = w. Hint: Use part (iii) of (C2). (ii) Let U = fC 2 GL2(R) : Cv = wg. With the help of (i), show that U is a left coset of Stab(v). (iii) Show that Stab(v) = fB??1AB : A 2 Stab(w)g where B is as in (i).