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Algebra 1. (a) State the definition of agroup. (b) State without proof whether or not the following are examples of groups i. The set Q' of non-zero rational numbers under multiplication ii. The set of symmetries of a cube under composition. iii. The set {-1,0,+1} under addition. (c) Prove that. a group cannot have more than one identity element [1+3+2=6 marks] 2. (a) State the definition of the dihedral group Dn (b) For each case below, prove that the two given groups are not isomorphic. i. S3 and Z6 ii. S4 and D12 iii. R+ and R' under multiplication (c) Prove that the symmetric group S. is not abelian for " 3. [1+3+2=6 marks] 3. (a) State the definition of the direct product G x H of two groups Gand H. (b) If ord(g) = 45, determine the order of 824. (c) Determine with proof whether z. x Z126 is isomorphic to z. x Zst [1+2+2=5 marks] 4. (a) State the definition of the cyclic subgroup (g). where g is an element of a group G. (b) Write down the elements of the cyclic subgroup ((2,4)) in z. x Zs and find the order of the generating element. (c) Let G be an abelian group. Prove that H = {g EGIg² = e} is a subgroup of G. [1+3+2=6 marks] 5. (a) State Lagrange's theorem. (b) Describe the left cosets of the subgroup Z in the group R. (c) Given a homomorphism f Z - Z20 such that f(1) = 14, determine (2013) and ker f- [1+2+2=5 marks] 6. (a) State the definition of a normal subgroup H of a group G. (b) Write down the left cosets and the right cosets of the subgroup H = (mu) in the dihedral group D2, where my represents reflection across a horizontal line. Write down the Cayley table for the quotient group D2/ H and determine which known group it is isomorphic to. Write down a homomorphism f D2 G whose kemel is H, where G is some group. (c) Prove the first isomorphism theorem, which states that if f G H a homomorphism, then G/ker f DE im f. [1+4+2=7 marks] Page of 2 Number Theory 7. (a) State the definition of the expression n m for n, € N. (b) Use the Euclidean algorithm to compute ged(30,50,75). Describe each step of the algorithm - the value of the greatest common divisor alone is not enough. (c) Compute ged(n,n + 1) form € N. [3+3+2 marks) 8. (a) State the Chinese reminder theorem for two factors (b) Consider the following system of congruences. x a (mod 30) @2 (mod 50) For al = 17, @2 = 27, find the smallest x € N that is a solution to the system. (c) For what values of € Z is it possible to solve the system of congruences in part (b)? [2 4 + marks] 9. (a) State the definition of the group Z; for pa prime. (b) Find the inverse of 315 in Z; (c) Findall x € Zi such that 16 Cx=0. [++3+3= marks) 10. (a) Let Z(i) be the set of Gaussian integers. Describe all Gaussian primes (real, purely imaginary, and complex). (b) For each of the following = € Z[ij, either prove that is a Gaussian prime or express z as a product of Gaussian primes. i. i ii. 7i iii. 13 (c) Assuming the properties of ged(21, =2) for Gaussian integers 21 and =2, prove the following statement: If p is a Gaussian prime and P zw then p zorp av. [2 2 10 marks]

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