= Question 5. [3+3=6 marks) Let S = {1,2,3,4} and G = S4. Write all of the elements of Gt and G() in each case. (a) T =

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Question 5 3 3 6 Marks Let S 1 2 3 4 And G S4 Write All Of The Elements Of Gt And G In Each Case A T 1 (78.93 KiB) Viewed 33 times

= Question 5. [3+3=6 marks) Let S = {1,2,3,4} and G = S4. Write all of the elements of Gt and G() in each case. (a) T = {1} (b) T = {1, 2, 3} Question 6. [4 marks) Find the elements in the alternating groups A3, and A4, respectively. Question 7. [3 marks) Find a subgroup of <Q,+> that contains Z but is different from both Z and Q. Question 8. [3 marks] Prove that if H and K are subgroups of a group G with operation *, then H K is a subgroup of G. Question 9. [3 marks Let H = {(1), (1 2)} and K = {(1), (1 2 3), (1 3 2)}. Both H and , K are subgroups of Sz. Show that HUK is not a subgroup of Sz. It follows that a union of subgroups is not necessarily a subgroup. а Question 10. [4 marks] Prove in detail that G(T) is a subgroup of G. Question 11. [3 marks) For G = S, state necessary and sufficient conditions on T, a subset of S = {1, 2, ...,n}, for G(T) = GT. =