R. Merris, On Burnside's Theorem, J. Algebra 48 (1977), 214-215. Math. Reviews 56 #5699. Research supported by NSF grant MCS 76-05946.

Suppose A is an (ordinary) irreducible representation of degree n of the finite group G. By Burnside's Theorem, {A(g): g in G} spans the vector space of n-by-n matrices. This article deals with the issue of choosing a subset S of G such that {A(g): g in S} is a basis for that vector space.

Theorem. Let M be the o(G)-by-o(G) matrix obtained by applying the character of the representation to each entry in the "multiplication" table for G. Then {A(g): g in S} is linearly independent if and only if the rows of M corresponding to the elements of S are linearly independent.

An alternative proof of the theorem was given by reviewer, R. Steinberg. A three-way generalization appeared in C. L. Morgan, On relations for representations of finite groups, Pacific J. Math. 78 (1978), 157-159.