# IIT JAM Mathematics Test Series 2023 : Group Theory – Homomorphism

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## IIT JAM Mathematics Test Series 2023 : Group Theory – Homomorphism

Q1. The number of group homomorphisms from the symmetric group S3 to the additive group Z/6Z is

(a) 1
(b) 2
(c) 3
(d) 0

Q2. Consider the two statements

A. Z6 is subgroup of Z12
B. Z5 is isomorphic to a subgroup of Z15

Then

(a) 1 is corect 2 is incorrect.
(b) 1 is incorrect 2 is correct.
(c) Both are correct.
(d) Both are incorrect.

Q3. The number of homomorphisms from Q8 to Z8

(a) 4
(b) 3
(c) 1
(d) None of these

Q4. The number of one-one homomorphisms from Z to S10 are

(a) 10
(b) 10!
(c) 2
(d) None of these

Q5. The group of one-one homomorphisms from Z21 to Z3 × Z7 is isomorphic to

(a) Z12
(b) U(12)
(c) Z21
(d) Z2 × Z6

Q6. The number of group homomorphisms from the symmetric group S3 to Z6 is

(a) 1
(b) 2
(c) 3
(d) 6

Q7. The number of non-isomorphic groups of order 10 is

(a) 1
(b) 2
(c) 3
(d) 4

Q8. Consider the group homomorphism φ : M2(R) → R given by φ(A) = trace(A). The kernel of φ is isomorphic to which of the following groups?

(a) M2(R)/H where H = {A ∈ M2(R) : φ(A) = 0}H
(b) R2
(c) R3
(d) GL(2,R)

Q9. The number of group homomorphism from Z3 to Z9 is

(a) 1
(b) 2
(c) 3
(d) 4

Q10. Let of Aut(G) automorphism of a group G. Which one of the following is NOT a cyclic group? denote the group

(a) Aut(Z4)
(b) Aut(Z6)
(c) Aut(Z8)
(d) Aut(Z10)

Q11. Let G = R\{0} and H = {-1,1} be groups under multiplication. Then the map φ:G → H defined by φ(x) = x/|x| is

(a) Not a homomorphism
(b) A one-one homomorphism, which is not onto
(c) An onto homomorphism, which is not one-one
(d) An isomorphism

Q12. Let G be a cyclic group of order 8, then its group of automorphisms has order

(a) 2
(b) 4
(c) 6
(d) 8

Q13. Let bijections f and g:R/{0,1} → R/{0,1} be defined by f(x) = 1/(1-x) and g(x) = x/(x-1) , and let G be the group generated by f and g under composition of mappings. It is-given that G has order 6. Then,

(a) G and its automorphisms group are both-Abelian
(b) G and its automorphisms group are both non-Abelian
(c) G is abelian but its automorphisms group is non-abelian
(d) G is non-abelian but its automorphisms group is Abelian

Q14. Let G and H be two groups. The groups G×H and H×G are isomorphic

(a) For any G and any H
(b) Only if one of them is cyclic
(c) Only if one of them is abelian
(d) Only if G and H are isomorphic

Q15. Let H = Z2 × Z6 and K = Z3 × Z4. Then

(a) H is isomorphic to K since both are cyclic
(b) H is not isomorphic to K since K is cyclic whereas H is not
(c) H is not isomorphic to K since there is no homomorphism from H to K
(d) None of these

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