# Who defined group representation in the modern way?

The modern definition of group representation is a homomorphism between a group $G$ and the group $GL(V, K)$ of some vector space over the field $K$.

But as far as I know, when Frobenius developed his character theory, he did not have this picture in his mind.

So, who invented (or discovered) this definition?

Frobenius did however give a more or less modern definition of a group representation shortly thereafter, also on Dedekind's suggestion:"Let $H$ be an abstract group, $A, B, C\dots$ be its elements. One associates to the element $A$ the matrix $(A)$, to the element $B$ the matrix $(B)$, etc., in such away that the group $\mathfrak{H}'$ is isomorphic to the group $\mathfrak{H}$, that is, $(A)(B)=(AB)$. Then I say that the substitutions or the matrices $(A),(B),(C)\dots$ represent the group $\mathfrak{H}$" ("isomorphic" did not yet mean one-to-one, so it corresponds to modern "homomorphic"). He then pointed out that the characters are given by the traces of matrices in "primitive" representations, defined as those with irreducible determinants.