Leftmultiplication

Left multiplication is the operation of multiplying an element on the left by a fixed element, in structures that support a binary product such as groups, rings, or algebras.

In group theory, for a group G and any g in G, the left multiplication map L_g: G → G is defined by L_g(h) = gh. The collection {L_g} forms the left regular action of G on itself. These maps satisfy L_e = identity and L_{gh} = L_g ∘ L_h, so the map g ↦ L_g is a group homomorphism from G into the symmetric group on G. The action is free and transitive, and G can be recovered from the action as the orbit of any element.

In ring theory, and more generally in algebras, left multiplication by a ∈ R is the map L_a: R → R given by L_a(x) = ax. It is an additive endomorphism and, when R is viewed as a left R-module, an R-linear operator. The assignment a ↦ L_a is a ring homomorphism from R to End(Z)(R), since L_{ab} = L_a ∘ L_b. Left multiplication by a is bijective precisely when a is a unit (has a two-sided inverse). More generally, L_a is injective if a is not a left zero-divisor, and surjective if a is right-invertible.

In linear algebra, left multiplication by a fixed matrix A acts on the space of matrices X by X ↦ AX, which is a linear operator. This perspective underpins several representation-theoretic constructions and reflects the way algebras act on themselves or on modules.