# Claim and proposed comparison

Work over Q. Let G = {1, g} with g^2 = 1. Let V and W each be the
one-dimensional vector space Q. Give V the trivial G-action and W the action
g(w) = -w.

Claim: V and W are isomorphic as G-representations.

Proposed proof: Both underlying vector spaces have dimension one, so choose a
linear isomorphism f: V -> W. Since the group action respects the vector space
structure, f is a G-equivariant isomorphism.