A representation of a group G is a homomorphism ρ: G → GL(V), where GL(V) is the group of invertible linear transformations on a vector space V. This translates the abstract algebraic structure of a group into concrete linear algebra, where powerful matrix techniques become available. The dimension of V is called the degree of the representation.
The central problem of group theory is understanding the structure of groups. One of the most powerful strategies for this is to represent abstract group elements as invertible linear maps on a vector space — that is, as matrices. A representation of a group G on a vector space V over a field F is a group homomorphism ρ: G → GL(V), where GL(V) denotes the group of all invertible linear transformations from V to itself. The homomorphism condition ρ(gh) = ρ(g)ρ(h) ensures that the group multiplication is faithfully reflected in matrix multiplication.
Why go through this translation? Because linear algebra is extraordinarily well-developed. We can diagonalize matrices, compute eigenvalues, take traces, and decompose spaces into direct sums. These operations have no direct analogues for abstract groups, but once we have a representation, we can apply all of linear algebra's machinery. A group that seemed opaque as a set with a binary operation becomes transparent when viewed through its action on a vector space.
The simplest example is the cyclic group ℤ/nℤ. A representation of this group on ℂ¹ is determined by where the generator 1 goes: it must map to a matrix (scalar) ζ with ζⁿ = 1, so ζ is an nth root of unity. Each root of unity gives a different one-dimensional representation. For a non-abelian example, the symmetric group S₃ has a two-dimensional representation where each permutation acts on ℝ² as a symmetry of an equilateral triangle — rotations and reflections become 2×2 matrices. This representation "sees" the geometric content of the group.
Two extreme cases frame the landscape. The trivial representation sends every element to the identity — it satisfies the homomorphism property vacuously but reveals nothing about G. The regular representation uses a vector space whose basis is indexed by the elements of G themselves, with each g ∈ G acting by permuting basis vectors. This representation is always faithful (injective) and contains every irreducible representation of G as a subrepresentation — a fact that makes it central to the structure theory you will develop in subsequent topics.