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Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE – (30 MARKS)
(a) Let G be a group, show that : (i) The identity element of G is unique. (3 Marks) (ii) Every ?????? has a unique inverse in G. (3 Marks)
(b) Express ? ? ? ? ? ? ? ? 897615432 987654321
as a product of disjoint cycles. (3 Marks)
(c) Write down the multiplication table of ??3 and determine the centre of ??3. (4 Marks) (d) Write down all the right cosets of ?? in ?? where ?? = ?? is a cycle group of order 10 and ?? = ??2 . (4 Marks) (e) Prove that H is a normal subgroup of ?? iff ??? ? ??, h ? ??, ??h??-1 ? ??. (5 Marks) (f) Determine whether the mapping Ø:?? ? ??, where G is the group of non-zero real numbers under multiplication and Ø is defined by Ø ?? = ??2 for all ?? ? ?? is a homeomorphism, if so find the Kernel of Ø. (4 Marks) (g) Determine the total number of inversions in the permutation. ? ? ? ?
? ? ? ?
523416 654321
(4 Marks)
2
QUESTION TWO – (20 MARKS)
(a) Given that ? ? ? ? ? ? ? ? ? 34512 54321 f , determine the orbits of ??. (4 Marks) (b) Determine whether the permutation ? ? ? ? ? ? 67845123 ???g is even or odd. (6 Marks)
(c) If G is a finite group whose order is a prime number p, show that G is cyclic. (5 Marks)
(d) Prove that if a group G is cyclic, then it is abelian. (5 Marks)
QUESTION THREE – (20MARKS)
(a) State and prove the Langrange’s theorem. (6 Marks)
(b) Let ?? = 1, 1,2 be a subgroup of ??3. Find all the left cosets of ?? in ??3. Determine the index of ?? in ??3 (5 Marks)
(c) (i) Let ?? and ?? be groups with identify element ?? and ?? respectively. If Ø:?? ? ?? is a homomorphism, show that Ø ?? = ?? and Ø ??-1 = Ø ?? -1 for all ?? ? ??. (4 Marks)
(ii) If Ø is a homomorphism from a group ?? into a group ?? with Kernel ??, show that ?? is a normal subgroup of ??. (5 Marks)
QUESTION FOUR – (20 MARKS)
(a) (i) Let G be a group and ?? ? ??. Define the centralizer ???? ?? of ?? ? ??. (2 Marks)
(ii) Show that ???? ?? is a subgroup of G. (6 Marks)
(b) Prove that if H is a normal subgroup of a group G and that the index of H in G is a prime number, then the quotient group ??/??, * is cyclic. (6 Marks)
(c) Let G be the set of all real 2 × 2 matrices ?? ?? ?? ??
, where ???? ? 0 under matrix
multiplication. Let
? ? ? ?
? ? ? ? ? ? ? ? ? ?
?
10 ba N . Prove that N is a normal subgroup of G. (6 Marks)
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