# Can every group be represented by a group of matrices?

Can every group be represented by a group of matrices?

Or are there any counterexamples? Is it possible to prove this from the group axioms?

Every finite group is isomorphic to a matrix group. This is a consequence of Cayley’s theorem: every group is isomorphic to a subgroup of its symmetry group. Since the symmetric group $S_n$ has a natural faithful permutation representation as the group of $n\times n$ 0-1 matrices with exactly one 1 in each row and column, it follows that every finite group is a matrix group.