Finitely generated abelian groups - structure theorems - One Line Questions
1.
What is the rank of the group G = Z_4 x Z_6? —
0
2.
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? —
0
3.
What is the rank of the group G = Z x Z_5? —
1
4.
What is the rank of the group G = Z_2 x Z_3 x Z_4? —
0
5.
Let G be a finitely generated abelian group. If G is isomorphic to Z_n, what is its rank? —
0
6.
Which of the following is NOT an invariant factor of Z_18? —
3
7.
What is the exponent of the group Z_2 x Z_4 x Z_6? —
12
8.
Consider the abelian group G = Z_2 x Z_3 x Z_4. What is the order of its torsion subgroup? —
24
9.
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? —
4
10.
Consider the group G = Z_4 x Z_6. What is the order of the torsion subgroup T(G)? —
24
11.
If G is a finitely generated abelian group and T(G) is its torsion subgroup, what is the structure of G/T(G)? —
A free abelian group.
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 product of unique prime powers.
13.
The structure theorem is crucial for classifying: —
All finitely generated abelian groups.
14.
The number of generators of a finitely generated abelian group is: —
Not necessarily unique.
15.
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: —
d_1 | d_2 | ... | d_k
16.
What is the elementary divisor decomposition of a finite abelian group? —
Direct product of cyclic groups of prime power order.
17.
Let G be a finitely generated abelian group. The torsion subgroup T(G) is unique. What about the free part F? —
F is unique.
18.
The structure theorem for finitely generated abelian groups is also known as the: —
Invariant Factor Decomposition Theorem.
19.
If a finitely generated abelian group G has no elements of finite order except the identity, what is its structure according to the theorem? —
G is isomorphic to Z^m for some m (a free abelian group of rank m).
20.
If a finitely generated abelian group G has rank 0, what does this imply about G? —
G is a finite group.
21.
According to the structure theorem, a finite abelian group is isomorphic to a direct product of: —
Finite cyclic groups only.
22.
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: —
Elementary divisor decomposition.
23.
The decomposition of a finitely generated abelian group into a direct product of cyclic groups is unique up to: —
Order of the factors.
24.
What is the fundamental property of a finitely generated abelian group that the structure theorem addresses? —
It is isomorphic to a direct product of cyclic groups.
25.
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? —
k > m
26.
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)? —
lcm(n_1, ..., n_k)
27.
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? —
Finite and infinite cyclic groups
28.
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: —
Both the order and the values of the prime powers.
29.
The structure theorem for finitely generated abelian groups is a direct consequence of the theory of: —
Modules over a principal ideal domain (PID).
30.
The structure theorem allows us to reduce the study of finitely generated abelian groups to the study of: —
Cyclic groups and free abelian groups.
31.
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? —
T(G) is isomorphic to a direct product of finite cyclic groups of prime power order.
32.
What is the fundamental theorem of finitely generated abelian groups primarily concerned with? —
The decomposition of finitely generated abelian groups into simpler, canonical forms.
33.
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? —
The rank of the torsion-free part of G.
34.
Let G be a finitely generated abelian group. The invariant factors are: —
The orders of the cyclic factors in the decomposition Z_{n_1} x ... x Z_{n_k}.
35.
The structure theorem for finitely generated abelian groups implies that any such group can be viewed as a module over which ring? —
The ring of integers Z.
36.
Which of the following is NOT a finitely generated abelian group? —
The set of all rational numbers under addition (Q).
37.
What is the 'rank' of a finitely generated abelian group G? —
The dimension of the vector space Q ⊗ G over Q.
38.
What is the 'torsion-free part' of a finitely generated abelian group G? —
The largest subgroup of G that is free abelian.
39.
What is the 'torsion subgroup' of a finitely generated abelian group G? —
The subgroup of elements of finite order.
40.
Consider the abelian group G = Z_12. Which of the following is a valid decomposition into cyclic groups? —
Z_4 x Z_3
41.
The invariant factor decomposition of Z_12 is Z_3 x Z_4. What is the elementary divisor decomposition? —
Z_2 x Z_2 x Z_3
42.
Consider the abelian group G = Z_2 x Z_2 x Z_4. What is its invariant factor decomposition? —
Z_4 x Z_4
43.
Let G = Z_2 x Z_4. What is its invariant factor decomposition? —
Z_8
44.
Let G = Z_2 x Z_4. What is its elementary divisor decomposition? —
Z_2 x Z_2 x Z_2
45.
Consider the abelian group G = Z_2 x Z_4 x Z_8. What is its elementary divisor decomposition? —
Z_2 x Z_2 x Z_2 x Z_2 x Z_2
46.
Consider the abelian group G = Z_3 x Z_3. What is its invariant factor decomposition? —
Z_3 x Z_3
47.
Consider the abelian group G = Z_3 x Z_3. What is its elementary divisor decomposition? —
Z_3 x Z_3
48.
Which of the following groups is NOT finitely generated? —
Q
49.
Consider the abelian group G = Z_6. What is its decomposition according to the structure theorem? —
Z_2 x Z_3
50.
Let G be a finitely generated abelian group. If G is torsion-free, it is isomorphic to: —
Z^m for some integer m.