Definition of groups - subgroups, homomorphism and isomorphism, group representations, irreducible representations, unitary representations - One Line Questions
1.
What is a homomorphism between two groups (G, *) and (G', ∘)? —
A function f: G -> G' such that f(a * b) = f(a) ∘ f(b) for all a, b in G.
2.
The concept of induced representations is used to construct representations of a group G from representations of: —
A subgroup H of G.
3.
What is a group representation? —
A mapping from a group G to the set of invertible matrices of a certain size, preserving the group structure.
4.
If D is a representation of G, and H is a subgroup of G, then the restriction of D to H, denoted Res<0xE1><0xB5><0x8F> D, is: —
A representation of H.
5.
What is an irreducible representation? —
A representation that cannot be decomposed into a direct sum of smaller representations.
6.
What is a unitary representation? —
A representation where D(g) is a unitary matrix for every g in G.
7.
What is a subgroup? —
A subset of a group that is itself a group under the same operation.
8.
What is the primary advantage of using characters instead of the full representation matrices? —
Characters are invariant under change of basis, making them unique for equivalent representations.
9.
Which property states that for any elements a, b, and c in a group G, (a * b) * c = a * (b * c)? —
Associative property
10.
For every element 'a' in a group G, there exists an element 'a⁻¹' such that a * a⁻¹ = a⁻¹ * a = e, where 'e' is the identity element. This is known as the: —
Inverse property
11.
Which property must a subset H of a group G satisfy to be a subgroup, besides being non-empty? —
All of the above.
12.
A representation D is equivalent to another representation D' if: —
There exists an invertible matrix S such that D'(g) = S⁻¹ D(g) S for all g in G.
13.
If D is a unitary representation, what is the relationship between D(g) and D(g)⁻¹? —
D(g)⁻¹ = D(g)* (conjugate transpose).
14.
In the context of group representations, what does it mean for a mapping D: G -> GL(n, C) to be a representation? —
D(g₁g₂) = D(g₁)D(g₂) for all g₁, g₂ in G, where D(g) are n x n matrices.
15.
In a group (G, *), what is the identity element 'e' characterized by? —
e * a = a for all a in G
16.
The condition for a mapping f: G -> G' to be a homomorphism is: —
f(a * b) = f(a) ∘ f(b)
17.
If f: G -> G' is a group homomorphism, what is the relationship between f(a⁻¹) and (f(a))⁻¹? —
f(a⁻¹) = (f(a))⁻¹.
18.
What is the closure property in the context of a group (G, *)? —
For any a, b in G, a * b is also in G.
19.
The set of all n x n unitary matrices forms a group under matrix multiplication. This group is denoted as: —
U(n)
20.
What is the defining characteristic of a subgroup H of G using the one-step subgroup test? —
H is non-empty and for all a, b in H, a * b⁻¹ is in H.
21.
Let H be a subset of a group G. Under what condition is H a subgroup of G? —
H is non-empty and for all a, b in H, a * b⁻¹ is in H.
22.
A homomorphism f: G -> G' is called an isomorphism if it is also: —
Bijective
23.
The theorem regarding the decomposition of a representation states that any finite-dimensional representation of a finite group (over the complex numbers) can be uniquely decomposed into a direct sum of: —
Irreducible representations.
24.
What is the relationship between isomorphism and group representations? —
Isomorphic groups have equivalent representation theories, meaning they have the same number and dimensions of irreducible representations.
25.
If D is a representation and H is a normal subgroup of G, the quotient group G/H can have its own representation induced from D. This is related to: —
Isomorphism theorems
26.
Consider the group S₃ (symmetric group on 3 elements) under composition. Which statement about its irreducible representations is true? —
It has exactly three irreducible representations.
27.
A representation D is reducible if: —
There exists a non-trivial invariant subspace under the action of the group.
28.
Consider the set of integers Z under addition. Is this set a group? —
Yes, it satisfies all group axioms.
29.
If for every pair of elements a, b in G, a * b = b * a, the group is called: —
Abelian or Commutative
30.
A set G with a binary operation * forms a group if it satisfies closure, associativity, existence of an identity element, and existence of inverse elements. What is this set of axioms called? —
Group Axioms
31.
The set of all 1x1 unitary matrices forms which group? —
U(1)
32.
If f: G -> G' is an isomorphism, then the kernel of f is: —
The identity element of G.
33.
The trivial subgroup of any group G is: —
The set containing only the identity element.
34.
If f: G -> G' is a group homomorphism, what is the image of the identity element of G under f? —
The identity element of G'.
35.
A representation D is called 'faithful' if: —
The kernel of the representation is only the identity element.
36.
What is the character of a group representation D? —
The trace of the matrix D(g).
37.
What is the fundamental property that defines a set G as a group under a binary operation *? —
There must exist an identity element and every element must have an inverse.
38.
What is the dimension of the trivial representation of any group G? —
1
39.
The number of non-equivalent irreducible representations of a finite group G is equal to: —
The number of conjugacy classes of G.
40.
The kernel of a group homomorphism f: G -> G' is defined as: —
The set of all elements in G whose image under f is the identity element of G'.
41.
What is the set GL(n, C) in the context of group representations? —
The set of all invertible n x n complex matrices.
42.
Which of the following is NOT a subgroup of the group of integers under addition (Z, +)? —
The set of odd integers (2Z+1, +)
43.
A group representation D: G -> GL(n, C) is called reducible if: —
There exists a proper non-trivial subspace of Cⁿ that is invariant under D(g) for all g ∈ G.
44.
Two groups G and G' are said to be isomorphic if: —
There exists a bijective homomorphism (isomorphism) between them.
45.
Why are unitary representations particularly important in physics? —
They preserve the inner product, ensuring probabilities remain constant in quantum mechanics.
46.
What does it mean for two groups to be isomorphic? —
They are structurally identical, differing only in the notation of their elements and operation.
47.
A matrix U is unitary if: —
U*U = I (identity matrix).
48.
If a group G has N irreducible representations, and the dimensions of these representations are d₁, d₂, ..., d<0xE2><0x82><0x99>, then which relation holds? —
Σᵢ<0xE1><0xB5><0x83>₁<0xE1><0xB5><0x83> dᵢ² = |G| (order of the group)
49.
Which of the following is NOT a property of the character χ of a representation D? —
χ(g⁻¹) = χ(g).