Download abstract algebra

BSc (Hons.) Mathematics
Cohort: BM/08/PT
Examinations for 2010 - 2011 / Semester 1
MODULE: ABSTRACT ALGEBRA
MODULE CODE: MATH 2124C
Duration: 2 Hours
Instructions to Candidates:
1. Answer ANY FOUR questions.
2. Questions may be answered in any order but your answers must show
the question number clearly.
3. Always start a new question on a fresh page.
4. All questions carry equal marks.
5. Total marks 100.
This question paper contains 5 questions and 4 pages.
Page 1 of 4
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ANSWER ANY FOUR QUESTIONS
QUESTION 1: (25 MARKS)
(a)
Suppose that A  1, 2, 3, 4, 5 and that  and  are permutations of A such
1 2 3 4 5 
1 2 3 4 5 
 and   
 .
that   
 4 2 5 3 1
 3 5 4 2 1
multiplication   in a similar form.
(b)
Express
the
permutation
(5 marks)
Let G and G be groups and let  : G  G  be a one-to-one function such
that  xy   x  y  for all x, y  G . Prove that  [G ] is a subgroup of G .
(7 marks)
(c)
Let H be a subgroup of G. Let the relation ~ L be defined on G by a ~ L b
if and only if a 1b  H . Prove that ~ L is an equivalence relation on G.
(8 marks)
(d)
Let H be a subgroup of a finite group G. Prove that the order of H is a
divisor of the order of G.
(5 marks)
QUESTION 2: (25 MARKS)
(a)
1 2 3 4 5 6 7 8 
 in S 8 . (5 marks)
Find the orbits of the permutation   
3 8 6 7 4 1 5 2
(b)
State the condition for which a permutation of a finite set is categorized as
even or odd.
(c)
Let H be a subgroup of a group G. Define the left coset of H containing an
element a.
(d)
(3 marks)
(2 marks)
Given that the group  6 is abelian. Find the partition of  6 into cosets of
the subgroup H  0, 3.
(4 marks)
P.T.O
Page 2 of 4
2010/S1
(e)
n
Let G1 , G2 ,..., Gn be groups. For a1 , a2 ,..., an  and b1 , b2 ,..., bn 
in
G
i
,
i 1
define a1 , a2 ,..., an  b1 , b2 ,..., bn  to be the element a1b1 , a2 b2 ,..., an bn  . Prove
n
that
G
i
is a group under this binary operation.
(11 marks)
i 1
QUESTION 3: (25 MARKS)
(a)
Prove that the finite indecomposable abelian groups are exactly the cyclic
groups with prime order power.
(b)
(7 marks)
Consider a map  of a group G onto a group G ' . Define the condition(s)
for  to be a homomorphism.
(c)
(2 marks)
Let  : G  G ' be a group homomorphism of G onto G ' . Prove that if G is
abelian then G ' is also abelian.
(d)
(6 marks)
Let  be a homomorphism of a group G onto a group G ' . Prove the
following three statements:
(i) If e is the identity element in G then  e  is the identity in G ' .
 
(ii) If a G then  a 1   (a) 1 .
(iii) If H is a subgroup of G then  H  is a subgroup of G ' .
(10 marks)
P.T.O
Page 3 of 4
2010/S1
QUESTION 4: (25 MARKS)
(a)
Let  : G  G ' be a homomorphism of a group G onto a group G ' . Define
the kernel of  .
(b)
(2 marks)
Let  : G  G ' be a group homomorphism and let H  ker  . Let a G .
Prove that the set  1  (a)  x  G /  ( x)   (a) is the left coset aH of H
and is also the right coset Ha of H.
(c)
(11 marks)
Let H be a subgroup of a group G. Prove that the left coset multiplication
is well defined by the equation (aH) (bH) = (ab)H if and only if H is a
normal subgroup of G.
(12 marks)
QUESTION 5: (25 MARKS)
(a)
Compute the factor group
4  6 / 2, 3
.
Show the necessary
workings.
(6 marks)
(b)
Let X be a set and G a group. Define an Action of G on X.
(4 marks)
(c)
Define a ring R, ,  with a set R and 2 binary operations + and  .
(5 marks)
(d)
If R is a ring with additive identity 0, then for any a, b   , prove the
following three results.
(i) 0a = a0 = 0.
(ii) a (-b) = (-a) b = -(ab).
(iii) (-a)(-b) = ab.
(10 marks)
***END OF QUESTION PAPER***
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2010/S1