Homomorphism, Automorphism and Cayley Theorem - Question Bank

1. Which property is essential for a function to be an automorphism?
A) It must map G to a proper subgroup of G.
B) It must be surjective only.
C) It must be bijective.
D) It must preserve order of elements but not necessarily the operation.
2. Cayley's Theorem implies that if a group G has order n, then G is isomorphic to a subgroup of:
A) S_n
B) A_n
C) Z_n
D) D_n
3. What is the image of the function f(x) = x^2 for G = {1, -1, i, -i}?
A) {1}
B) {-1}
C) {1, -1}
D) {1, -1, i, -i}
4. Consider the group G = {1, -1, i, -i} under multiplication. The function f: G -> G defined by f(x) = x^2 is:
A) An automorphism
B) A homomorphism but not an automorphism
C) An isomorphism but not an automorphism
D) Not a homomorphism
5. If G is a group and H is a subgroup, when is H necessarily the kernel of some homomorphism from G?
A) Always
B) If H is the center of G
C) If H is a normal subgroup of G
D) If G/H is isomorphic to a subgroup of Sym(G)
6. The order of the automorphism group of S_3 is:
A) 6
B) 12
C) 24
D) 36
7. What is the kernel of the homomorphism f(x) = 2x for G = (R, +)?
A) R
B) {1}
C) {0}
D) {-1, 1}
8. Let G = (R, +) be the group of real numbers under addition. The function f(x) = 2x is a:
A) Homomorphism but not an automorphism
B) Automorphism
C) Isomorphism but not an automorphism
D) Neither a homomorphism nor an isomorphism
9. If f: G -> H is a homomorphism, and f(g) = e_H for all g in G, then f is the:
A) Identity homomorphism
B) Trivial homomorphism
C) Inclusion homomorphism
D) Zero homomorphism
10. Cayley's Theorem shows that every group is a 'sub-object' of what type of more concrete structure?
A) Vector spaces
B) Rings
C) Permutation groups
D) Fields
11. The center of a group G, denoted Z(G), is invariant under which type of transformation?
A) Any homomorphism
B) Any isomorphism
C) Any inner automorphism
D) Any outer automorphism
12. The set of elements in G that commute with all other elements of G forms a subgroup called the:
A) Normalizer
B) Centralizer
C) Center
D) Stabilizer
13. Let G be a group. An automorphism f of G is an isomorphism from G to:
A) A subgroup of G
B) The trivial group
C) G itself
D) A quotient group of G
14. Which property is preserved by an automorphism but not necessarily by a mere homomorphism?
A) Identity element mapping
B) Inverse mapping
C) Order of elements
D) Group structure (isomorphism)
15. If f: G -> H is a homomorphism, and K = Ker(f), then G/K is isomorphic to Im(f). This is a statement of:
A) Cayley's Theorem
B) The First Isomorphism Theorem
C) The Second Isomorphism Theorem
D) The concept of kernel
16. What is the order of the group of automorphisms of the cyclic group Z_3?
A) 1
B) 2
C) 3
D) 6
17. Let G be a group. The set of all homomorphisms from G to G is called the:
A) Automorphism group
B) Endomorphism monoid
C) Isomorphism group
D) Center of G
18. If f: G -> H and g: H -> K are group homomorphisms, then the composition g o f: G -> K is:
A) Always an isomorphism
B) Always an automorphism
C) Always a homomorphism
D) Not necessarily a homomorphism
19. Cayley's Theorem guarantees that a group of order n is isomorphic to a subgroup of S_n, where S_n is the symmetric group on n elements. The order of S_n is:
A) n
B) n!
C) 2n
D) n^2
20. Consider the group S_3 (symmetric group on 3 elements). Which of the following is an automorphism of S_3?
A) Mapping every element to the identity.
B) Mapping every element to its inverse.
C) Mapping every element to itself (identity map).
D) Mapping transposition (1 2) to (1 3).
21. An automorphism that fixes every element of the group G is called the:
A) Trivial automorphism
B) Inner automorphism
C) Outer automorphism
D) Identity automorphism
22. Which of the following is NOT necessarily true for a group homomorphism f: G -> H?
A) f preserves subgroups: if S is a subgroup of G, then f(S) is a subgroup of H.
B) f preserves subgroups: if T is a subgroup of H, then f^-1(T) is a subgroup of G.
C) f maps the identity of G to the identity of H.
D) f maps inverses to inverses: f(g^-1) = (f(g))^-1.
23. Let f: G -> H be a homomorphism. If f is injective (a monomorphism), then its kernel must be:
A) G
B) The identity element {e_G}
C) H
D) The trivial subgroup {e_H}
24. This fundamental theorem relates a homomorphism, its kernel, and its image.
A) Lagrange's Theorem
B) Sylow's Theorems
C) First Isomorphism Theorem
D) Second Isomorphism Theorem
25. If f: G -> H is a homomorphism, and K is the kernel of f, then G/K is isomorphic to:
A) H
B) Im(f)
C) G
D) Ker(f)
26. The kernel of the map that sends g to the inner automorphism I_g(x) = gxg^-1 is the:
A) Center of G
B) Commutator subgroup of G
C) Derived subgroup of G
D) Automorphism group of G
27. The set of all inner automorphisms of G forms a subgroup of the group of all automorphisms, denoted as:
A) Inn(G)
B) Aut(G)
C) Ker(G)
D) Im(G)
28. An inner automorphism of a group G is an automorphism of the form:
A) f(x) = axa^-1 for a fixed a in G
B) f(x) = x
C) f(x) = x^-1
D) f(x) = axa for a fixed a in G
29. The set of all automorphisms of a group G forms a group under the operation of:
A) Addition
B) Multiplication
C) Function composition
D) Element-wise operation
30. Let G = Z_n (integers modulo n under addition). Which function is an automorphism of G?
A) f(x) = x + 1 (mod n)
B) f(x) = 2x (mod n)
C) f(x) = -x (mod n)
D) f(x) = x^2 (mod n)
31. Which property does an automorphism preserve that a general homomorphism might not?
A) The operation
B) The identity element
C) The structure of the group in its entirety
D) The inverse property
32. If f: G -> H is a group homomorphism, and H is abelian, is the image of f necessarily abelian?
A) No, the image can be non-abelian.
B) Yes, the image is always abelian.
C) Only if f is a monomorphism.
D) Only if G is abelian.
33. If f: G -> H is a group homomorphism, and G is abelian, is the image of f necessarily abelian?
A) No, the image can be non-abelian.
B) Yes, the image is always abelian.
C) Only if f is an epimorphism.
D) Only if G is a cyclic group.
34. Cayley's Theorem is significant because it shows that any abstract group can be realized as a group of:
A) Matrices
B) Permutations
C) Functions
D) Vectors
35. Let G be a group and g in G. The left multiplication map L_g: G -> G is defined by L_g(x) = gx. Is L_g a homomorphism?
A) No, it does not preserve the operation.
B) Yes, L_g(xy) = g(xy) = (gx)y = L_g(x)y, which is not L_g(x)L_g(y).
C) Yes, L_g(xy) = g(xy) = (gx)(gy) = L_g(x)L_g(y).
D) No, it is only defined for the identity element.
36. The specific type of homomorphism used in Cayley's Theorem to represent elements of G as permutations is typically:
A) Right multiplication
B) Left multiplication
C) Inverse mapping
D) Identity mapping
37. In the context of Cayley's Theorem, the group of permutations that G is isomorphic to is denoted by:
A) Sym(G)
B) Aut(G)
C) S_G
D) Inj(G)
38. Cayley's Theorem states that every group G is isomorphic to a subgroup of the group of:
A) Symmetric polynomials
B) Permutations on the set G
C) Real numbers under addition
D) Integers under multiplication
39. If f: G -> H is an isomorphism, then the order of an element g in G is equal to the order of which element in H?
A) e_H
B) f(g)
C) g
D) f(e_G)
40. The image of a group homomorphism is always a(n):
A) Normal subgroup of the domain G
B) Subgroup of the domain G
C) Subgroup of the codomain H
D) Normal subgroup of the codomain H
41. The kernel of a group homomorphism is always a(n):
A) Subgroup of the image
B) Subgroup of the codomain H
C) Normal subgroup of the domain G
D) Normal subgroup of the codomain H
42. The image of a group homomorphism f: G -> H is defined as:
A) {g in G | f(g) = e_H}
B) {h in H | exists g in G such that f(g) = h}
C) {g in G | f(g) = e_G}
D) {h in H | f(h) = e_H}
43. The kernel of a group homomorphism f: G -> H is defined as:
A) {h in H | f(g) = h for some g in G}
B) {g in G | f(g) = e_H}
C) {g in G | f(g) = e_G}
D) {h in H | f(h) = e_H}
44. Which of the following is NOT a property of a group homomorphism f: G -> H?
A) f(e_G) = e_H
B) f(g^-1) = (f(g))^-1
C) f(g^n) = (f(g))^n for any integer n
D) f(g1 * g2) = f(g1) + f(g2)
45. An isomorphism from a group G to itself is called a(n):
A) Homomorphism
B) Endomorphism
C) Automorphism
D) Isomorphism
46. A group homomorphism f: G -> H that is both a monomorphism and an epimorphism is called a:
A) Endomorphism
B) Automorphism
C) Isomorphism
D) Homomorphism
47. A group homomorphism f: G -> H is called a monomorphism if it is:
A) One-to-one and onto
B) Onto
C) One-to-one
D) Neither one-to-one nor onto
48. A group homomorphism f: G -> H is called an epimorphism if it is:
A) One-to-one and onto
B) Onto
C) One-to-one
D) Neither one-to-one nor onto
49. Let f: G -> H be a group homomorphism. If e_G is the identity in G and e_H is the identity in H, which property must f satisfy?
A) f(g1 * g2) = f(g1) + f(g2)
B) f(g1 * g2) = f(g1) * f(g2)
C) f(g1 * g2) = f(g1) - f(g2)
D) f(g1 * g2) = f(g2) * f(g1)
50. What is the fundamental property that a homomorphism preserves between two algebraic structures?
A) The number of elements
B) The binary operation
C) The identity element
D) The order of elements
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