Homomorphism, Automorphism and Cayley Theorem - One Line Questions
1.
What is the image of the function f(x) = x^2 for G = {1, -1, i, -i}? —
{1, -1}
2.
The image of a group homomorphism f: G -> H is defined as: —
{h in H | exists g in G such that f(g) = h}
3.
The kernel of a group homomorphism f: G -> H is defined as: —
{g in G | f(g) = e_H}
4.
What is the order of the group of automorphisms of the cyclic group Z_3? —
2
5.
The order of the automorphism group of S_3 is: —
6
6.
Let G be a group. An automorphism f of G is an isomorphism from G to: —
G itself
7.
The set of all automorphisms of a group G forms a group under the operation of: —
Function composition
8.
If G is a group and H is a subgroup, when is H necessarily the kernel of some homomorphism from G? —
If H is a normal subgroup of G
9.
If f: G -> H and g: H -> K are group homomorphisms, then the composition g o f: G -> K is: —
Always a homomorphism
10.
Consider the group G = {1, -1, i, -i} under multiplication. The function f: G -> G defined by f(x) = x^2 is: —
A homomorphism but not an automorphism
11.
The center of a group G, denoted Z(G), is invariant under which type of transformation? —
Any inner automorphism
12.
Let G be a group. The set of all homomorphisms from G to G is called the: —
Endomorphism monoid
13.
If f: G -> H is a homomorphism, and K = Ker(f), then G/K is isomorphic to Im(f). This is a statement of: —
The First Isomorphism Theorem
14.
The kernel of the map that sends g to the inner automorphism I_g(x) = gxg^-1 is the: —
Center of G
15.
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? —
f(g)
16.
A group homomorphism f: G -> H that is both a monomorphism and an epimorphism is called a: —
Isomorphism
17.
Which of the following is NOT necessarily true for a group homomorphism f: G -> H? —
f preserves subgroups: if T is a subgroup of H, then f^-1(T) is a subgroup of G.
18.
Which of the following is NOT a property of a group homomorphism f: G -> H? —
f(g1 * g2) = f(g1) + f(g2)
19.
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? —
f(g1 * g2) = f(g1) * f(g2)
20.
An inner automorphism of a group G is an automorphism of the form: —
f(x) = axa^-1 for a fixed a in G
21.
Let G = Z_n (integers modulo n under addition). Which function is an automorphism of G? —
f(x) = -x (mod n)
22.
Let f: G -> H be a homomorphism. If f is injective (a monomorphism), then its kernel must be: —
The identity element {e_G}
23.
If f: G -> H is a homomorphism, and K is the kernel of f, then G/K is isomorphic to: —
Im(f)
24.
An isomorphism from a group G to itself is called a(n): —
Automorphism
25.
Let G = (R, +) be the group of real numbers under addition. The function f(x) = 2x is a: —
Homomorphism but not an automorphism
26.
Which property is preserved by an automorphism but not necessarily by a mere homomorphism? —
Order of elements
27.
If f: G -> H is a homomorphism, and f(g) = e_H for all g in G, then f is the: —
Trivial homomorphism
28.
The set of all inner automorphisms of G forms a subgroup of the group of all automorphisms, denoted as: —
Inn(G)
29.
Which property is essential for a function to be an automorphism? —
It must be bijective.
30.
This fundamental theorem relates a homomorphism, its kernel, and its image. —
First Isomorphism Theorem
31.
Consider the group S_3 (symmetric group on 3 elements). Which of the following is an automorphism of S_3? —
Mapping every element to itself (identity map).
32.
Cayley's Theorem is significant because it shows that any abstract group can be realized as a group of: —
Permutations
33.
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: —
n!
34.
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? —
Yes, L_g(xy) = g(xy) = (gx)(gy) = L_g(x)L_g(y).
35.
If f: G -> H is a group homomorphism, and G is abelian, is the image of f necessarily abelian? —
Yes, the image is always abelian.
36.
If f: G -> H is a group homomorphism, and H is abelian, is the image of f necessarily abelian? —
Yes, the image is always abelian.
37.
The image of a group homomorphism is always a(n): —
Subgroup of the codomain H
38.
The set of elements in G that commute with all other elements of G forms a subgroup called the: —
Center
39.
A group homomorphism f: G -> H is called an epimorphism if it is: —
Onto
40.
A group homomorphism f: G -> H is called a monomorphism if it is: —
One-to-one
41.
What is the kernel of the homomorphism f(x) = 2x for G = (R, +)? —
{0}
42.
The specific type of homomorphism used in Cayley's Theorem to represent elements of G as permutations is typically: —
Left multiplication
43.
Cayley's Theorem implies that if a group G has order n, then G is isomorphic to a subgroup of: —
S_n
44.
The kernel of a group homomorphism is always a(n): —
Normal subgroup of the domain G
45.
In the context of Cayley's Theorem, the group of permutations that G is isomorphic to is denoted by: —
Sym(G)
46.
Cayley's Theorem states that every group G is isomorphic to a subgroup of the group of: —
Permutations on the set G
47.
What is the fundamental property that a homomorphism preserves between two algebraic structures? —
The binary operation
48.
Which property does an automorphism preserve that a general homomorphism might not? —
The structure of the group in its entirety
49.
An automorphism that fixes every element of the group G is called the: —
Identity automorphism
50.
Cayley's Theorem shows that every group is a 'sub-object' of what type of more concrete structure? —
Permutation groups