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.
However, there are infinite groups which are not matrix groups, for example, the symmetric group on an infinite set or the metaplectic group.
Note that every group can be represented non-faithfully by a group of matrices: just take the trivial representation. My answer above is for the question of whether every group has a faithful matrix representation.