Practice Questions on Homomorphism with Answers

**Questions on Homomorphism with Answers**

**Q1. The number of group homomorphisms from the symmetric group S _{3} to the additive group ^{Z}/_{6Z} is**

(a) 1

(b) 2

(c) 3

(d) 0

**Q2. Consider the two statements**

**A. Z _{6} is subgroup of Z_{12}**

B. Z_{5} is isomorphic to a subgroup of Z_{15}

**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 Q _{8} to Z_{8}**

(a) 4

(b) 3

(c) 1

(d) None of these

**Q4. The number of one-one homomorphisms from Z to S _{10} are**

(a) 10

(b) 10!

(c) 2

(d) None of these

**Q5. The group of one-one homomorphisms from Z _{21} to Z_{3} × Z_{7} is isomorphic to**

(a) Z_{12}

(b) U(12)

(c) Z_{21}

(d) Z_{2} × Z_{6}

**Q6. The number of group homomorphisms from the symmetric group S _{3} to Z_{6} 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 : M _{2}(R) → R given by φ(A) = trace(A). The kernel of φ is isomorphic to which of the following groups?**

(a) M_{2}(R)/H where H = {A ∈ M_{2}(R) : φ(A) = 0}H

(b) R_{2}

(c) R_{3}

(d) GL(2,R)

**Q9. The number of group homomorphism from Z _{3} to Z_{9} 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(Z_{4})

(b) Aut(Z_{6})

(c) Aut(Z_{8})

(d) Aut(Z_{10})

**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 = Z _{2} × Z_{6} and K = Z_{3} × Z_{4}. 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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*Answer Key*

*Answer Key*

01. | (b) | 06. | (b) | 11. | (c) |

02. | (b) | 07. | (b) | 12. | (b) |

03. | (a) | 08. | (c) | 13. | (b) |

04. | (d) | 09. | (c) | 14. | (a) |

05. | (d) | 10. | (c) | 15. | (b) |