ALGEBRA PROBLEM SHEET 1. Let f and g be the following permutation...

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ALGEBRA PROBLEM SHEET 1. Let f and g be the following permutations in the symmetric group S10, written in two-line notation. 􏰀 1 2 3 4 5 6 7 8 9 10 􏰁 􏰀 1 2 3 4 5 6 7 8 9 10 􏰁 f= g= 7 1 8 9 3 10 4 5 2 6 4 5 9 1 3 8 7 6 2 10 (a) Write down and clearly label the following eight permutations in disjoint cycle notation. Briefly explain why your answer for f 2222 is correct. f g f◦g g◦f f−1 f2 f3 f2222 2. For each case below, prove that the two given groups are not isomorphic. (a) D15 and Z30 (b) S5 and S6 (c) D360 and S6 3. (c) List the elements of the group Z24 and determine the order of each element. 4. Determine with proof whether the following functions f : G → H are isomorphisms. (a) f (x) = √x, where G = R+ under multiplication and H = R+ under multiplication 5. For each case below, prove that the two given groups are isomorphic. (a) R∗undermultiplicationandR∗undertheoperationx∗y=3xy 8. LetGbethegroupRunderadditionandletHbethegroupR+undermultiplication.Supposethatf:G → H is an isomorphism such that f (4) = 4. (a) Determine f (8), f (20), f (0), f (2) and f (1)

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