Further Math Ib Ppq's on Srg
1. (a) Let A be the set of all 2 × 2 matrices of the form , where a and b are real numbers, and a2 + b2 0. Prove that A is a group under matrix multiplication.
(10)
(b) Show that the set: M = forms a group under matrix multiplication.
(5)
(c) Can M have a subgroup of order 3? Justify your answer.
(2)
(Total 17 marks)
3. (a) Define an isomorphism between two groups (G, o) and (H, •).
(2)
(b) Let e and e be the identity elements of groups G and H respectively.
Let f be an isomorphism between these two groups. Prove that f(e) = e.
(4)
(c) Prove that an isomorphism maps a finite cyclic group onto another finite cyclic group.
(4)
(Total 8 marks)
4. Consider the set U = {1, 3, 5, 9, 11, 13} under the operation *, where * is multiplication modulo 14. (In all parts of this problem, the general properties of multiplication modulo n may be assumed.)
(a) Show that (3 * 9) * 13 = 3 * (9 * 13).
(2)
(b) Show that (U, *) is a group.
(11)
(c) (i) Define a cyclic group.
(2)
(ii) Show that (U, *) is cyclic and find all its generators.
(7)
(d) Show that there are only two non-trivial proper subgroups of this group, and find them.
(7)
(Total 29 marks)
5. Consider a group (G, o) with identity e. Suppose that H is a subset of G such that H = {x G : x o a = a o x, for all a G}. Show that (H, o) is a subgroup of (G, o), by showing that
(a) e H;
(2)
(b) if x, y H, then x o y H,
[i.e. show that (x o y) o a = a o (x o y)];
(5)
(c) if x H, then x–1 H.
(4)
(Total 11 marks)
6. Let X and Y be two non-empty sets.
(a) Define the operation X Y by X Y = (X Y) u (X F).
Prove that (X Y) = (X Y) (X Y).
(3)
(b) Let f : be defined by f(n) = n + 1, for all n . Determine if f is an injection, a surjection, or a bijection. Give reasons for your answer.
(3)
(c) Let h : X Y, and let R be an equivalence relation on Y. y1Ry2 denotes that two elements y1 and y2 of Y are related.
(10)
(b) Show that the set: M = forms a group under matrix multiplication.
(5)
(c) Can M have a subgroup of order 3? Justify your answer.
(2)
(Total 17 marks)
3. (a) Define an isomorphism between two groups (G, o) and (H, •).
(2)
(b) Let e and e be the identity elements of groups G and H respectively.
Let f be an isomorphism between these two groups. Prove that f(e) = e.
(4)
(c) Prove that an isomorphism maps a finite cyclic group onto another finite cyclic group.
(4)
(Total 8 marks)
4. Consider the set U = {1, 3, 5, 9, 11, 13} under the operation *, where * is multiplication modulo 14. (In all parts of this problem, the general properties of multiplication modulo n may be assumed.)
(a) Show that (3 * 9) * 13 = 3 * (9 * 13).
(2)
(b) Show that (U, *) is a group.
(11)
(c) (i) Define a cyclic group.
(2)
(ii) Show that (U, *) is cyclic and find all its generators.
(7)
(d) Show that there are only two non-trivial proper subgroups of this group, and find them.
(7)
(Total 29 marks)
5. Consider a group (G, o) with identity e. Suppose that H is a subset of G such that H = {x G : x o a = a o x, for all a G}. Show that (H, o) is a subgroup of (G, o), by showing that
(a) e H;
(2)
(b) if x, y H, then x o y H,
[i.e. show that (x o y) o a = a o (x o y)];
(5)
(c) if x H, then x–1 H.
(4)
(Total 11 marks)
6. Let X and Y be two non-empty sets.
(a) Define the operation X Y by X Y = (X Y) u (X F).
Prove that (X Y) = (X Y) (X Y).
(3)
(b) Let f : be defined by f(n) = n + 1, for all n . Determine if f is an injection, a surjection, or a bijection. Give reasons for your answer.
(3)
(c) Let h : X Y, and let R be an equivalence relation on Y. y1Ry2 denotes that two elements y1 and y2 of Y are related.