A student defines a representation of a group G as any function ρ: G → GL(V). What critical condition is missing?
Aρ must be injective
Bρ must be a group homomorphism, i.e., ρ(gh) = ρ(g)ρ(h) for all g, h ∈ G
Cρ must be surjective onto GL(V)
DV must be finite-dimensional
A representation must preserve the group operation: ρ(gh) = ρ(g)ρ(h). Without this homomorphism condition, the map would assign matrices to group elements arbitrarily, losing all structural information. Injectivity is not required (the trivial representation sends every element to the identity matrix and is perfectly valid). Surjectivity is also not required, and V can be infinite-dimensional in general.
Question 2 True / False
Every group has at least one representation.
TTrue
FFalse
Answer: True
Every group has the trivial representation, which sends every group element to the identity matrix on a one-dimensional space: ρ(g) = 1 for all g ∈ G. This is clearly a homomorphism since ρ(gh) = 1 = 1·1 = ρ(g)ρ(h). While it carries no information about the group's structure, it is a valid representation. More interestingly, Cayley's theorem guarantees every group has a faithful (injective) representation via permutation matrices.
Question 3 Short Answer
The trivial representation sends every group element to the identity transformation. Why is this still considered a legitimate representation despite conveying no structural information about G?
Think about your answer, then reveal below.
Model answer: It satisfies the definition: it is a homomorphism from G to GL(V), since ρ(gh) = I = I·I = ρ(g)ρ(h) for all g, h ∈ G. Representations are defined by the homomorphism property, not by how much information they carry.
The trivial representation is the kernel-maximal extreme — its kernel is all of G. At the other extreme, a faithful representation has trivial kernel. Both are valid homomorphisms. The trivial representation plays a role analogous to the zero function in analysis: structurally degenerate but necessary for the theory to be clean (e.g., it appears as a summand in decompositions).
Question 4 Multiple Choice
If G is a finite group of order n, what is the degree of the representation obtained from the left regular action of G on the vector space with basis indexed by elements of G?
A1
Bn − 1
Cn
Dn²
The left regular representation uses a vector space with one basis vector for each group element, so its dimension equals |G| = n. Each group element g acts by permuting the basis vectors via left multiplication: g·eₕ = e_{gh}. This gives an n-dimensional representation that is always faithful — distinct group elements produce distinct permutation matrices.