,
,This paper consists of 3 pages.
INSTRUCTIONS
Answer all the questions.
[TURN OVER]
, 2 MAT3702
October/November 2024
QUESTION 1
(a) Define a relation ∼ on |R by x ∼ y if |x| = |y| and check as to whether ∼ is an equivalence relation
on |R or not (5)
(b) Let G be a group and show that for any a ∈ G, there is a unique b ∈ G such that ab = e = ba,
where e ∈ G is the identity element in G (5)
(c) Let G be an abelian group with H, K as subgroups of G and prove that HK = {hk | h ∈ H, k ∈ K}
is a subgroup of G (5)
(d) Find all generators of ZZ 8 (5)
(e) Let G be a finite group with H, K as its subgroups. Then prove that the order of H ∩ K is a
common divisor of the order of H and the order of K (5)
[25]
QUESTION 2
(a) Let R∗ be the multiplicative group of nonzero real numbers and R∗∗ be the multiplicative group
of positive real numbers. Define f : R∗ −→ R∗∗ by f (x) = x2 and check as to whether f is a
hommomorphism or not (5)
[Hint: first check as to whether f is well-defined or not]
(b) Compute each product (6)
(i) (1 2)(5 3 2 1 4)(2 3)
(ii) (1 2 3 4)(2 3 4 5)
(c) Let |R ∗ be the group of nonzero real numbers under multiplication and check as to whether ϕ :
|R ∗ −→ |R ∗ defined by ϕ(x) = |x| is a homomorphism or not and if it is a homomorphism, then
find its kernel (7)
[Hint: first check as to whether ϕ is well-defined or not]
(d) Let G be a group and N be a normal subgroup of G. Then show that π : G −→ G/N defined by
π(x) = N x for x ∈ G is a surjective homomorphisnm and find its kernel (7)
[Hint: first check as to whether π is well-defined or not]
[25]
[TURN OVER]
,This paper consists of 3 pages.
INSTRUCTIONS
Answer all the questions.
[TURN OVER]
, 2 MAT3702
October/November 2024
QUESTION 1
(a) Define a relation ∼ on |R by x ∼ y if |x| = |y| and check as to whether ∼ is an equivalence relation
on |R or not (5)
(b) Let G be a group and show that for any a ∈ G, there is a unique b ∈ G such that ab = e = ba,
where e ∈ G is the identity element in G (5)
(c) Let G be an abelian group with H, K as subgroups of G and prove that HK = {hk | h ∈ H, k ∈ K}
is a subgroup of G (5)
(d) Find all generators of ZZ 8 (5)
(e) Let G be a finite group with H, K as its subgroups. Then prove that the order of H ∩ K is a
common divisor of the order of H and the order of K (5)
[25]
QUESTION 2
(a) Let R∗ be the multiplicative group of nonzero real numbers and R∗∗ be the multiplicative group
of positive real numbers. Define f : R∗ −→ R∗∗ by f (x) = x2 and check as to whether f is a
hommomorphism or not (5)
[Hint: first check as to whether f is well-defined or not]
(b) Compute each product (6)
(i) (1 2)(5 3 2 1 4)(2 3)
(ii) (1 2 3 4)(2 3 4 5)
(c) Let |R ∗ be the group of nonzero real numbers under multiplication and check as to whether ϕ :
|R ∗ −→ |R ∗ defined by ϕ(x) = |x| is a homomorphism or not and if it is a homomorphism, then
find its kernel (7)
[Hint: first check as to whether ϕ is well-defined or not]
(d) Let G be a group and N be a normal subgroup of G. Then show that π : G −→ G/N defined by
π(x) = N x for x ∈ G is a surjective homomorphisnm and find its kernel (7)
[Hint: first check as to whether π is well-defined or not]
[25]
[TURN OVER]
