Although nonabelian groups were originally studied as certain sets of permutations, the notion of an abstract group was soon introduced in order to cover a much wider variety of examples. Cayley's theorem, which states that every group is isomorphic to a group of permutations, shows that the abstract definition does not really include additional classes of groups. Cayley's theorem also shows that any group can be ‘represented’ in a concrete fashion as a set of permutations.

A representation of a group may be useful even if it is not a one-to-one function. This is illustrated by the proofs of Cauchy's theorem and the Sylow theorems, which represent a group G as sets of permutations on various sets associated with the given group. Although permutation representations have been used very effectively, other representations have proved to be more important. Even before the study of permutation groups, group characters were used in working with finite abelian groups. The initial ideas can be traced to work of Gauss, but the definition of a character of an abelian group as a homomorphism from the group into the multiplicative group of nonzero complex numbers was not given until 1897 (by Dedekind).

The most useful representations of a finite group appear to be as groups of matrices. Major contributions to the initial development were made by Probenius, at the end of the nineteenth century. The matrix groups were assumed to be subgroups of GLn(C), and in 1897 Frobenius formulated the modern definition of a group character as the trace of a matrix representation.