Further Math Ib Ppq's on Srg

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  • Topic: Group, Modular arithmetic, Integer
  • Pages : 16 (3373 words )
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  • Published : January 12, 2011
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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.
Define a relation S on X by the following:
For all a, b  X, a S b if and only if h(a) R h(b).
Determine if S is an equivalence relation on X.
(4)
(Total 10 marks)

7.(a)Let f1, f2, f3, f4 be functions defined on – {0}, the set of rational numbers excluding zero, such that f1(z) = z, f2(z) = –z, f3(z) = , and f4(z) = –, where z  – {0}.
Let T= {f1, f2, f3, f4}. Define  as the composition of functions i.e. (f1  f2)(z) = fl(f2(z)).Prove that (T, ) is an Abelian group. (6)
(b)Let G = {1, 3, 5, 7} and (G, ) be the multiplicative group under the binary operation , multiplication modulo 8. Prove that the two groups (T, ) and (G, ) are isomorphic. (5)
(Total 11 marks)

8.Let a, b and p be elements of a group (H, *) with an identity element e. (a)If element a has order n and element a–l has order m, then prove that m = n. (5)
(b)If b=p–1 *a*p, prove, by mathematical induction, that bm = p–1 * am * p, where m = 1, 2,.... (4)
(Total 9 marks)

9.A–B is the set of all elements that belong to A but not to B. (a)Use Venn diagrams to verify that (A – B)  (B – A) = (A  B) – (A  B). (2)
(b)Use De Morgan’s laws to prove that (A – B)  (B – A) = (A  B) – (A  B). (4)
(Total 6 marks)

10.Show that the set H = forms a group under matrix multiplication. (You may assume that matrix multiplication is associative).
(Total 6 marks)

11.(a)State Lagrange’s theorem.
(b)Let (G, °) be a group of order 24 with identity element e. Let a  G, and suppose that a12  e and a8  e. Prove that (G, °) is a cyclic group with generator a.
(Total 7 marks)

forms a group under matrix multiplication.
(You may assume that matrix multiplication is associative).
(Total 6 marks)

12.Let S =
(a)Prove that S is a group under multiplication, ×, of numbers. (b)For x = a + , define f(x) = a – . Prove that f is an isomorphism from (S, ×) onto (S, ×).
(Total 11 marks)

13.Let S = {(x, y)  x, y  }, and let (a, b), (c, d)  S. Define the relation  on S as...
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