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.
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(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.