Q 1 Inthefollowing,determinewhetherthesystemsdescribedaregroups. If they are not groups, point out which axioms fail to hold. (a) (Z,?), a?b = a?b (b) (Q,?), a?b = ab (c) (2Z,?), a?b = a + b (d) (N,?), a?b = a + b

Q2 Checkifthefollowingaregroups. Iftheyaregroups(includechecking closure), show if they are abelian, non-abelian, ?nite or in?nite. (a) nZ: multipliers of n by integers under addition. (b) Zn : integer modulo n under addition. Q3 (a) Make addition table forZ5. (b) Find the orders of elements ofZ5. Q4 Let ?,?,µ ? S6 such that ? =1 2 3 4 5 6 3 1 4 5 6 2, ? =1 2 3 4 5 6 2 4 1 3 6 5, and µ =1 2 3 4 5 6 5 2 4 3 1 6. Compute the indicated products. ??,?2?,µ?2,??1??

MAT311-Tutorial 2

Question 1 (a) Let G be a group and let a,b ? G. Show that i. (ab)2 = a2b2 if and only if G is abelian. ii. (ab)?1 = a?1b?1 if and only if G is abelian. (b) In S3,giveanexampleoftwoelements a,b suchthat (ab)2 6= a2b2. (c) Is S3 abelian? Give a reason to your answer. (d) Find the orders of elements of S3. Argue why ??1 1 = ?1.

Question 2 (a) For D3 = {e,r,r2,s,rs,r2s}, verify that (rs)r2 = r2s, (r2s)(rs) = r and sr2 = rs with r3 = e = s2 and sr = r2s. (b) Work out clearly (r3s)(r2s) and (r2s)(r3s), the elements of D4. (c) Find the order of r3 and r2s in D4.

Question 3 (a) Find the cycle decomposition of the permutation ? =1 2 3 4 5 6 7 8 9 10 3 6 5 7 8 9 10 1 2 4, (b) Express the permutation µ ? S10 as a product of transpositions. (c) Find the order of µ Question 4 (a) Let G be a group and let a,x ? G. Prove that (xax?1)n = xanx?1 for n ? 1. Deduce further that xax?1 and a have the same order. (b) Show that the element A =1 1 0 1? GL(2,IR) is of in?nite order.