Finitely generated abelian groups - structure theorems - Question Bank

1. The structure theorem states that any finitely generated abelian group G is isomorphic to Z_{p_1^{a_1}} x ... x Z_{p_k^{a_k}} x Z^m, where p_i are primes and a_i are positive integers. This is known as the:
A) Invariant factor decomposition.
B) Elementary divisor decomposition.
C) Torsion-free decomposition.
D) Rank decomposition.
2. Consider the abelian group G = Z_2 x Z_2 x Z_4. What is its invariant factor decomposition?
A) Z_2 x Z_2 x Z_4
B) Z_4 x Z_4
C) Z_8
D) Z_2 x Z_8
3. The invariant factor decomposition of Z_12 is Z_3 x Z_4. What is the elementary divisor decomposition?
A) Z_12
B) Z_3 x Z_4
C) Z_2 x Z_2 x Z_3
D) Z_6 x Z_2
4. What is the fundamental theorem of finitely generated abelian groups primarily concerned with?
A) The classification of all groups.
B) The structure of groups based on generators and relations.
C) The decomposition of finitely generated abelian groups into simpler, canonical forms.
D) The properties of non-abelian groups.
5. Let G be a finitely generated abelian group. If G is isomorphic to Z_n, what is its rank?
A) 0
B) 1
C) n
D) undefined
6. The structure theorem allows us to reduce the study of finitely generated abelian groups to the study of:
A) Simple groups.
B) Cyclic groups of prime order.
C) Cyclic groups and free abelian groups.
D) Non-abelian groups.
7. If G is a finite abelian group, its exponent is the smallest positive integer n such that nx = 0 for all x in G. What is the exponent of Z_2 x Z_4?
A) 2
B) 4
C) 6
D) 8
8. The invariant factor decomposition of Z_{n_1} x ... x Z_{n_k} is Z_{d_1} x ... x Z_{d_m} where d_1 | ... | d_m. What is the relationship between k and m?
A) k = m
B) k > m
C) k < m
D) No fixed relationship.
9. What is the rank of the group G = Z_2 x Z_3 x Z_4?
A) 0
B) 1
C) 2
D) 3
10. Consider the abelian group G = Z_2 x Z_3 x Z_4. What is the order of its torsion subgroup?
A) 2
B) 3
C) 4
D) 24
11. Which of the following is NOT a finitely generated abelian group?
A) The set of all rational numbers under addition (Q).
B) The set of all integers under addition (Z).
C) The set of all integers modulo n under addition (Z_n).
D) The direct product Z x Z.
12. The structure theorem is analogous to the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be uniquely factored into:
A) A sum of primes.
B) A product of unique prime numbers.
C) A product of unique prime powers.
D) A sum of powers of 2.
13. What is the rank of the group G = Z x Z_5?
A) 0
B) 1
C) 2
D) 5
14. If a finitely generated abelian group G has rank 0, what does this imply about G?
A) G is isomorphic to Z.
B) G is a finite group.
C) G is an infinite cyclic group.
D) G is trivial.
15. The structure theorem is crucial for classifying:
A) All finite groups.
B) All simple groups.
C) All finitely generated abelian groups.
D) All non-abelian groups.
16. Let G be a finitely generated abelian group. The torsion subgroup T(G) is unique. What about the free part F?
A) F is unique.
B) F is unique up to isomorphism.
C) F is not necessarily unique.
D) F is always Z.
17. Consider the abelian group G = Z_3 x Z_3. What is its elementary divisor decomposition?
A) Z_3 x Z_3
B) Z_9
C) Z_3
D) Z
18. Consider the abelian group G = Z_3 x Z_3. What is its invariant factor decomposition?
A) Z_3 x Z_3
B) Z_9
C) Z_3
D) Z
19. What is the exponent of the group Z_2 x Z_4 x Z_6?
A) 2
B) 4
C) 6
D) 12
20. If G is a finitely generated abelian group and G ≅ Z_{n_1} x ... x Z_{n_k} x Z^m, what is the exponent of G (if G is finite)?
A) m
B) lcm(n_1, ..., n_k)
C) n_1 * ... * n_k
D) k
21. The structure theorem for finitely generated abelian groups implies that any such group can be viewed as a module over which ring?
A) The ring of integers Z.
B) The field of rational numbers Q.
C) The ring of polynomials Z[x].
D) The field of real numbers R.
22. Let G = Z_2 x Z_4. What is its elementary divisor decomposition?
A) Z_2 x Z_4
B) Z_8
C) Z_2 x Z_2 x Z_2
D) Z_4 x Z_2
23. Let G = Z_2 x Z_4. What is its invariant factor decomposition?
A) Z_2 x Z_4
B) Z_8
C) Z_2 x Z_2
D) Z_4 x Z_2
24. The rank of a finitely generated abelian group G is equal to the dimension of the vector space Q ⊗ G over Q. What is the rank of Z_3 x Z_5?
A) 0
B) 1
C) 2
D) 3
25. Which of the following is NOT an invariant factor of Z_18?
A) 18
B) 9
C) 2
D) 3
26. The invariant factor decomposition of a finite abelian group G is of the form Z_{d_1} x Z_{d_2} x ... x Z_{d_k} where:
A) d_1 | d_2 | ... | d_k
B) d_k | d_{k-1} | ... | d_1
C) d_1, d_2, ..., d_k are prime powers.
D) d_1 = d_2 = ... = d_k.
27. Consider the abelian group G = Z_2 x Z_4 x Z_8. What is its elementary divisor decomposition?
A) Z_2 x Z_4 x Z_8
B) Z_2 x Z_2 x Z_2 x Z_4
C) Z_2 x Z_2 x Z_2 x Z_2 x Z_2
D) Z_16
28. What is the elementary divisor decomposition of a finite abelian group?
A) Direct product of cyclic groups of any order.
B) Direct product of cyclic groups of prime power order.
C) Direct product of infinite cyclic groups.
D) Direct product of free abelian groups.
29. For a finite abelian group G, the structure theorem states it is isomorphic to a direct product of cyclic groups of prime power order. This decomposition is unique up to:
A) Order of the factors.
B) The specific prime powers.
C) Isomorphism.
D) Both the order and the values of the prime powers.
30. Let G be a finitely generated abelian group. The invariant factors are:
A) The orders of the cyclic factors in the decomposition Z_{n_1} x ... x Z_{n_k}.
B) The prime powers that divide the order of the group.
C) The number of generators.
D) The order of the torsion subgroup.
31. The structure theorem for finitely generated abelian groups is a direct consequence of the theory of:
A) Rings.
B) Modules over a principal ideal domain (PID).
C) Fields.
D) Vector spaces.
32. What is the rank of the group G = Z_4 x Z_6?
A) 0
B) 1
C) 2
D) 4
33. Consider the group G = Z_4 x Z_6. What is the order of the torsion subgroup T(G)?
A) 4
B) 6
C) 12
D) 24
34. The number of generators of a finitely generated abelian group is:
A) Always unique.
B) Not necessarily unique.
C) Always equal to the rank.
D) Always equal to the order of the group.
35. Let G be a finitely generated abelian group. If G is torsion-free, it is isomorphic to:
A) Z_n for some n.
B) Z.
C) Z^m for some integer m.
D) A direct product of finite cyclic groups.
36. The decomposition of a finitely generated abelian group into a direct product of cyclic groups is unique up to:
A) Isomorphism.
B) Order of the factors.
C) Both the order and the values of the factors.
D) The number of factors.
37. Which of the following groups is NOT finitely generated?
A) Z_5
B) Z x Z
C) Q
D) Z_2 x Z_3
38. The structure theorem can be stated as G ≅ T(G) x F, where T(G) is the torsion subgroup and F is a free abelian group. What is the structure of T(G) itself?
A) T(G) is always isomorphic to Z.
B) T(G) is always a free abelian group.
C) T(G) is isomorphic to a direct product of finite cyclic groups of prime power order.
D) T(G) is isomorphic to Z_n for some n.
39. If G is a finitely generated abelian group and T(G) is its torsion subgroup, what is the structure of G/T(G)?
A) A finite cyclic group.
B) A finite abelian group.
C) A free abelian group.
D) A trivial group.
40. What is the 'torsion-free part' of a finitely generated abelian group G?
A) The subgroup consisting of all elements of G.
B) The subgroup consisting of elements of finite order.
C) The largest subgroup of G that is free abelian.
D) The quotient group G / T(G), where T(G) is the torsion subgroup.
41. Consider the abelian group G = Z_12. Which of the following is a valid decomposition into cyclic groups?
A) Z_12
B) Z_4 x Z_3
C) Z_6 x Z_2
D) Z_2 x Z_2 x Z_3
42. Consider the abelian group G = Z_6. What is its decomposition according to the structure theorem?
A) Z_6
B) Z_2 x Z_3
C) Z
D) Z_2 x Z_2
43. The structure theorem for finitely generated abelian groups is also known as the:
A) Fundamental Theorem of Arithmetic.
B) Cayley-Hamilton Theorem.
C) Invariant Factor Decomposition Theorem.
D) Sylow Theorems.
44. According to the structure theorem, a finite abelian group is isomorphic to a direct product of:
A) Infinite cyclic groups only.
B) Finite cyclic groups only.
C) Both finite and infinite cyclic groups.
D) Free abelian groups.
45. What is the 'rank' of a finitely generated abelian group G?
A) The smallest number of generators for G.
B) The number of elements in G.
C) The dimension of the vector space Q ⊗ G over Q.
D) The order of the torsion subgroup of G.
46. If a finitely generated abelian group G has no elements of finite order except the identity, what is its structure according to the theorem?
A) G is isomorphic to Z_n for some n.
B) G is isomorphic to Z.
C) G is isomorphic to Z^m for some m (a free abelian group of rank m).
D) G is trivial (contains only the identity element).
47. What is the 'torsion subgroup' of a finitely generated abelian group G?
A) The subgroup of elements of finite order.
B) The subgroup of elements of infinite order.
C) The subgroup generated by all elements.
D) The subgroup of elements with prime order.
48. Let G be a finitely generated abelian group. The structure theorem guarantees that G is isomorphic to Z_{n_1} x Z_{n_2} x ... x Z_{n_k} x Z^m, where Z_n represents the cyclic group of order n and Z represents the infinite cyclic group. What does 'm' represent in this decomposition?
A) The number of generators of G.
B) The rank of the torsion-free part of G.
C) The exponent of G.
D) The number of distinct prime factors of the orders of torsion elements.
49. The structure theorem for finitely generated abelian groups states that any such group G is isomorphic to a direct product of which types of groups?
A) Only finite cyclic groups
B) Only infinite cyclic groups
C) Finite and infinite cyclic groups
D) Free abelian groups of finite rank
50. What is the fundamental property of a finitely generated abelian group that the structure theorem addresses?
A) It is always cyclic.
B) It is isomorphic to a direct product of cyclic groups.
C) It is isomorphic to a free group.
D) Its order is always finite.
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