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A simple proof of the group-theoretic Zhang-Yeung inequality
Abstract
Zhang and Yeung (IEEE Trans. Inf. Theory, 1998) established the first non-Shannon-type inequality that holds for all entropic vectors. Chan and Yeung (IEEE Trans. Inf. Theory, 2002) showed that there is a one-to-one relation between linear entropy inequalities and multiplicative inequalities involving cardinalities of subgroups of a finite group. The Shannon inequality admits a simple proof in the group theoretic setting, but a direct proof of the translation of Zhang--Yeung's inequality to the group-theoretic setting remained elusive. We resolve this open problem, giving a direct proof of the group-theoretic Zhang--Yeung inequality for cardinalities of subgroups of finite groups, using elementary group-theoretic and counting arguments.
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BibTeX
@misc{nieuwboer2026simple,
title = {{A simple proof of the group-theoretic Zhang-Yeung inequality}},
author = {Harold Nieuwboer and Lubashan Pathirana},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35451},
eprint = {2609.35451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35451},
}
RIS
TY - UNPB
TI - A simple proof of the group-theoretic Zhang-Yeung inequality
AU - Harold Nieuwboer
AU - Lubashan Pathirana
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35451
UR - https://arxiv.org/abs/2609.35451
AB - Zhang and Yeung (IEEE Trans. Inf. Theory, 1998) established the first non-Shannon-type inequality that holds for all entropic vectors. Chan and Yeung (IEEE Trans. Inf. Theory, 2002) showed that there is a one-to-one relation between linear entropy inequalities and multiplicative inequalities involving cardinalities of subgroups of a finite group. The Shannon inequality admits a simple proof in the group theoretic setting, but a direct proof of the translation of Zhang--Yeung's inequality to the group-theoretic setting remained elusive. We resolve this open problem, giving a direct proof of the group-theoretic Zhang--Yeung inequality for cardinalities of subgroups of finite groups, using elementary group-theoretic and counting arguments.
ER -