Finite-dimensional representation obstruction from elementary abelian subgroups
= Finite-dimensional representation obstruction from elementary abelian subgroups
Commuting complex involutions are simultaneously diagonalizable, so $(C_2)^n$ can act faithfully in dimension $d$ only if $n\leq d$. A group containing $(C_2)^n$ for arbitrarily large $n$, such as the full permutation group of $\mathbb N$, has no faithful finite-dimensional complex representation.