This paper studies the one-way communication complexity of the subgroup
membership problem, a classical problem closely related to basic questions in
quantum computing. Here Alice receives, as input, a subgroup H of a finite
group G; Bob receives an element x∈G. Alice is permitted to send a
single message to Bob, after which he must decide if his input x is an
element of H. We prove the following upper bounds on the classical
communication complexity of this problem in the bounded-error setting: (1) The
problem can be solved with O(log∣G∣) communication, provided the subgroup
H is normal; (2) The problem can be solved with O(dmax⋅log∣G∣)
communication, where dmax is the maximum of the dimensions of the
irreducible complex representations of G; (3) For any prime p not dividing
∣G∣, the problem can be solved with O(dmax⋅logp) communication,
where dmax is the maximum of the dimensions of the irreducible
\F_p-representations of G