Normal Subgroups and Quotient Groups - One Line Questions
1.
Let G = ℤ₆ and N = {0, 3}. Which of the following is a coset of N in G? —
{1, 4}
2.
What is the order of the quotient group G/N if G is a finite group and N is a normal subgroup of G? —
|G| / |N|
3.
If G/N is isomorphic to H, and N is a normal subgroup of G, what is the relationship between the orders of these groups? —
|G| = |N| * |H|
4.
Let G = GL₂(ℝ) and N = {A ∈ G | det(A) = 1} = SL₂(ℝ). What is the order of the quotient group G/N? —
∞
5.
What is the order of the quotient group ℤ₁₀ / {0, 5}? —
5
6.
What is the order of the quotient group D₄ / Z(D₄), where Z(D₄) is the center of D₄? —
4
7.
What is the order of the quotient group ℤ₁₂ / {0, 6}? —
6
8.
Let G = ℤ₁₅ and N = {0, 3, 6, 9, 12}. What is the order of the quotient group G/N? —
3
9.
If N is a normal subgroup of G, and K is a subgroup of G such that N ⊆ K, when is K/N a normal subgroup of G/N? —
Always
10.
If N is a normal subgroup of G, then for any two elements aN and bN in G/N, their product is defined as (aN)(bN) = ? —
(ab)N
11.
Let G = ℤ/nℤ. Which subgroup is normal? —
All subgroups are normal because ℤ/nℤ is abelian.
12.
If N is a normal subgroup of G, then the set of left cosets G/N forms a group under the operation (aN)(bN) = (ab)N. What is this group called? —
Both Quotient Group and Factor Group
13.
If G is a group and N is a normal subgroup, the elements of G/N are: —
Cosets of N in G.
14.
Which of the following is a consequence of N being a normal subgroup of G? —
Every left coset is equal to its corresponding right coset.
15.
What is the fundamental property that defines a normal subgroup N of a group G? —
N is a subgroup of G and for every g in G, gNg⁻¹ = N.
16.
What is the kernel of the natural projection homomorphism π: G → G/N, where N is a normal subgroup of G? —
N
17.
Consider a group homomorphism φ: G → H. The image of φ, Im(φ), is always isomorphic to which quotient group? —
G / Ker(φ)
18.
The set of all cosets of a normal subgroup N in G forms a group under the operation (aN)(bN) = (ab)N. This operation is well-defined if and only if: —
N is a normal subgroup.
19.
Which condition is NOT equivalent to N being a normal subgroup of G? —
gNg⁻¹ ⊆ N for all g ∈ G
20.
The set of all normal subgroups of a group G forms a: —
Lattice under intersection and generation
21.
The Second Isomorphism Theorem states that if N is a normal subgroup of G and K is a subgroup of G, then (NK)/N is isomorphic to K/(N ∩ K). This implies that if N ⊆ K, then K/N is isomorphic to: —
K/(N ∩ K)
22.
The First Isomorphism Theorem for groups states that if φ: G → H is a surjective group homomorphism, then G/Ker(φ) is isomorphic to which group? —
H
23.
If φ: G → H is a group homomorphism, what is the relationship between the kernel of φ, Ker(φ), and normal subgroups? —
Ker(φ) is always a normal subgroup of G.
24.
Which theorem directly relates normal subgroups to group isomorphisms? —
The First Isomorphism Theorem
25.
If N is a normal subgroup of G, and G/N is isomorphic to the trivial group {e}, what can be concluded about N? —
N = G
26.
If G is a non-abelian simple group, what are its normal subgroups? —
The trivial subgroup {e} and G itself.
27.
If G is an abelian group, what can be said about all of its subgroups? —
Every subgroup is normal.
28.
What is the identity element in the quotient group G/N? —
The identity coset, eN = N
29.
If G is a group and N is a normal subgroup, the order of an element gN in the quotient group G/N is: —
The smallest positive integer k such that (gN)ᵏ = N.
30.
What is the commutator subgroup G' of a group G? —
The subgroup generated by all commutators xyx⁻¹y⁻¹ for x, y ∈ G.
31.
The Third Isomorphism Theorem states that if N and K are normal subgroups of G with N ⊆ K, then G/K is isomorphic to (G/N)/(K/N). This theorem is useful for understanding: —
The structure of quotient groups of quotient groups.
32.
Consider the group of integers under addition, (ℤ, +). Which subgroup is normal? —
All of the above
33.
Which of the following is NOT always a normal subgroup of a group G? —
The commutator subgroup G'
34.
If G is a simple group, what are its only normal subgroups? —
The trivial subgroup {e} and G itself.
35.
The center of a group G, Z(G) = {z ∈ G | zg = gz for all g ∈ G}, is always a normal subgroup of G. True or False? —
True
36.
Is the commutator subgroup G' always a normal subgroup of G? —
Yes
37.
If G/N is a cyclic group, does it imply that G is cyclic? —
No, for example, G = D₄ and N = Z(D₄).
38.
If N is a normal subgroup of G, and K is a normal subgroup of N, is K necessarily a normal subgroup of G? —
No, for example, let G = S₄, N = A₄, K = V₄ (Klein four-group).
39.
Let G = S₃ and N = A₃ = {e, (123), (132)}. Is N a normal subgroup of S₃? —
Yes, because A₃ has index 2 in S₃.
40.
Let G = S₄ and N = A₄. Is N a normal subgroup of G? —
Yes, because A₄ has index 2 in S₄, and subgroups of index 2 are always normal.
41.
Let G = GL₂(ℝ) (the general linear group of 2x2 invertible matrices with real entries) and N = SL₂(ℝ) (the special linear group). Is N a normal subgroup of G? —
Yes, because for any A in G and B in N, det(ABA⁻¹) = det(A)det(B)det(A⁻¹) = 1 * det(B) * 1 = det(B) = 1, so ABA⁻¹ is in N.
42.
Consider the group of quaternions Q₈ = {±1, ±i, ±j, ±k}. Is the subgroup H = {±1} a normal subgroup of Q₈? —
No, because iHi⁻¹ = {i(1)i⁻¹, i(-1)i⁻¹} = {1, -1} = H, but jHj⁻¹ = {j(1)j⁻¹, j(-1)j⁻¹} = {1, -1} = H, and kHk⁻¹ = {k(1)k⁻¹, k(-1)k⁻¹} = {1, -1} = H, so it IS normal.
43.
Let G = S₃ (the symmetric group on 3 elements) and N = {e, (12)}. Is N a normal subgroup of S₃? —
No, because (13)N(13)⁻¹ = {(13)e(13)⁻¹, (13)(12)(13)⁻¹} = {e, (23)} ≠ N.
44.
Let G = D₃ (dihedral group of order 6) and N = {e, r, r²} (rotations). Is N a normal subgroup of D₃? —
Yes, because N is a subgroup of index 2.
45.
Let G = D₄ (the dihedral group of order 8) and N = {e, r², s, sr²} where r is rotation and s is reflection. Is N a normal subgroup of D₄? —
Yes, because N is the center of D₄.
46.
Let G = A₄ (the alternating group on 4 elements) and N = {e, (12)(34), (13)(24), (14)(23)}. Is N a normal subgroup of A₄? —
Yes, because N is the center of A₄.
47.
Consider the group of non-zero rational numbers under multiplication (ℚ*, ×). Let N = {1, -1}. Is N a normal subgroup of ℚ*? —
Yes, because for any a ∈ ℚ*, a(1)a⁻¹ = 1 and a(-1)a⁻¹ = -1, so aNa⁻¹ = N.
48.
Let G = ℤ × ℤ and N = {(k, 0) | k ∈ ℤ}. Is N a normal subgroup of G? —
Yes, because ℤ × ℤ is abelian.
49.
Let G = ℤ₁₂ (the cyclic group of order 12) and N = {0, 4, 8}. Is N a normal subgroup of ℤ₁₂? —
Yes, because ℤ₁₂ is abelian, so all its subgroups are normal.
50.
Consider the group of integers under addition (ℤ, +). The subgroup 5ℤ is normal. What is the quotient group ℤ/5ℤ isomorphic to? —
ℤ₅