Abstract Questions

 Which of the operations depicted by the following three multiplication tables defined on the set G = {a, b, c, d} form a group. Give reasons for your conclusion. (i) ◦ a b c d a a c d a b b b c d c c d a b d d a b c (ii) ◦ a b c d a a b c d b b c d a c c d a b d d a b c (iii) ◦ a b c d a a b c d b b a c d c c b a d d d d b c 2. (a) Write out the Cayley table for the group U(12). (b) Is U(12) isomorphic to either, both, or none of U(10) and/or U(8)? Explain your answer. 3. Prove the following theorem about exponentiation in groups. Theorem. For all g, h ∈ G, (a) (g m) n = g mn for all m, n ∈ Z. (b) (gh) −1 = h −1 g −1 and by extension (gh) n = (h −1 g −1 ) −n for all n ∈ Z. (c) Furthermore, if G is abelian, then (gh) n = g nh n for all n ∈ Z. 4. Find a specific example of a group G and elements g, h ∈ G where (gh) n 6= g nh n for some n 6= 0 or 1. Show that your group and the two elements you chose satisfy the criteria. 5. Prove that for n > 2, there is an element k ∈ U(n), k 6= 1, where k 2 = 1. 7. Prove or disprove: SL(2,Z), the set of 2 × 2 integer entry matrices of determinant 1, is a subgroup of SL(2, R). page 1 out of 2 6. Let G be a group in which (ab) 2 = a 2 b 2 for all a and b in G. Prove that G is abelian. 8. Prove or disprove: If H and K are subgroups of a group G, then H ∪ K is a subgroup of G. Recall: Defintion. For a group G with identity e, (i) An element g ∈ G is of finite order if for some n ∈ Z + , g n = e. (ii) The order of such an element g as in (i) is the smallest positive power n such that g n = e. 9. (a) Find all the orders of the elements in Z × 7 by filling in the table below for each power of the element until you get 1, the identity of Z × 7 . A couple of lines are already filled in. k z }| { 1 2 3 4 5 6 order 1 k 1 1 2 k 3 k 4 k 4 2 1 3 5 k 6 k (b) Based on the above table, is Z × 7 a cyclic group or not? Include reasons for your answer. 10. (i) Explain why Z 6=0 9 is not a group under 9 . (ii) Even though Z 6=0 9 is not a group, find which elements in Z 6=0 9 are invertible and find their orders. 11. Aff(Z4 ) = all invertible 2 × 2 matrices over Z4 of the form  a b 0 1 . The operation of Aff(Z4 ) is matrix multiplication using the ⊕4 and 4 to combine the entries when multiplying matrices. List all the elements of Aff(Z4 ) wit