The One-Way Communication Complexity of Group Membership

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 $x \in G$. 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 $O(d_{\max} \cdot \log |G|)$ communication, where $d_{\max}$ 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 $O(d_{\max} \cdot \log p)$ communication, where $d_{\max}$ is the maximum of the dimensions of the irreducible $\F_p$-representations of $G$.

💡 Research Summary

The paper investigates the one‑way (Alice‑to‑Bob) communication complexity of the subgroup‑membership problem, a fundamental decision problem that asks whether a given element x of a finite group G belongs to a subgroup H supplied to Alice. In the model, Alice may send a single message to Bob, after which Bob must output the correct answer with bounded error (typically error ≤ 1/3). While the two‑way communication complexity of this problem is well understood, the one‑way setting imposes a severe restriction and had not been studied in depth. The authors provide three upper‑bound results that connect the communication cost to structural properties of the group and to representation‑theoretic parameters.

Result 1 – Normal subgroups.
If H is a normal subgroup of G, the authors show that O(log |G|) bits suffice. The key observation is that the quotient group G/H inherits the group operation, so membership of x in H is equivalent to checking whether the coset xH is the identity coset. Alice can encode a succinct description of H (for example, a generating set or a hash of the normal closure) using O(log |G|) bits. Because the normality guarantees that the coset operation commutes with the group operation, Bob can verify membership without any additional interaction. This matches the trivial lower bound and demonstrates that the one‑way model incurs no extra cost for normal subgroups.

Result 2 – General subgroups via complex representations.
For arbitrary subgroups, the authors turn to the representation theory of G over the complex numbers. Let ρ range over all irreducible complex representations of G, with dimensions d_ρ, and let d_max = max_ρ d_ρ. For each ρ, the image ρ(H) is a subspace of the matrix algebra M_{d_ρ}(ℂ). By Schur’s Lemma, ρ(H) is either the whole space or a proper subalgebra, and the relationship between ρ(H) and ρ(x) determines whether x lies in H. Alice therefore sends, for each ρ, a compressed description of ρ(H) that requires O(d_ρ·log |G|) bits (e.g., a basis of the subspace or a short fingerprint). Summing over all representations yields a total communication cost of O(d_max·log |G|). This bound is tight up to the factor d_max and improves dramatically over the naïve O(|G|) bound whenever the largest irreducible dimension grows sub‑linearly (as is the case for many families of groups, such as the symmetric groups where d_max = n‑1).

Result 3 – Modular representations for primes coprime to |G|.
When a prime p does not divide |G|, the group algebra ℤ_p