1

Does every group, given for example by its multiplication table, have a matrix representation? If not, how does one proof for a given group that no matrix representation exists?

In addition, as for groups whose elements $g$ can be represented by a matrix $D(g)$ the representation is defined by

$$ D(g_1)D(g_2) = D(g_1g_2) $$

what would be the analogue condition defining non-matrix representations?

Dilaton
  • 1,177

1 Answers1

1

Every permutation on $n$ elements (that is, every element of $S_n$) can be represented by an $n{\,\times\,}n$ matrix

$\qquad$https://en.wikipedia.org/wiki/Permutation_matrix

and every finite group is isomorphic to a subgroup of $S_n$, where $n=|G|$.

$\qquad$https://en.wikipedia.org/wiki/Cayley%27s_theorem

Therefore every finite group has a matrix representaton.

quasi
  • 58,772