Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 3 / 6 / c

Set $n=6$ in part (b). The quotient $W=u^{\perp }/⟨u⟩$ has dimension four and a nondegenerate alternating form. The coordinate action therefore definesProve it is faithful before comparing orders. If a permutation lies in the kernel of a group homomorphism, it fixes the quotient class of each two-element subset $A$. That class has exactly the two representatives $A$ and $A+\Omega =\Omega \setminus A$. Their cardinalities are two and four. A coordinate permutation preserves cardinality, so it must fix $A$ itself, not exchange it with its complement. Thus it fixes every two-element subset. Intersecting the fixed sets $\{i,j\}$ and $\{i,k\}$ shows it fixes each singleton $\{i\}$, so it is the identity. Hence $\rho$ is injective.

By part (a),An injection between finite groups of equal order is surjective. Choosing a symplectic basis of $W$ identifies the form with the standard one, soThis realizes the exceptional isomorphism through a concrete binary permutation module. Equality of orders alone would not establish it; the faithful form-preserving action supplies the required group homomorphism.