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.

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.
There are many groups which have infinite number of elements. Then of course you can't represent them with a finite bunch of stored matrices. For example the group of 2D rotations over real valued 2D vectors. But you can sometimes get arbitrarily close by using only one or a few matrices and natural numbers as exponents.

$$R_\theta = \begin{bmatrix}\cos(\theta)&amp;\sin(\theta)\\-\sin(\theta)&amp;\cos(\theta)\end{bmatrix}$$

Now choose $\theta = \epsilon = 2\pi\cdot 2^{-n}&gt;0$

$${R_\theta} ^{2^n} = I$$

Given some measure of closeness, we can go on to prove we can get arbitrarily close to any element in the group if we just allow $n$ and matrix-exponents (integers) be large enough. In the example above we have in some sense a "resolution" of $n$ bits. But it's real convenient to just have to consider one "building block" $R_\theta$ and it's exponents and not try to store all possible elements separately. 

( We would quickly run out of memory if we tried for any infinite group )