On Group Violations of Inequalities in five Subgroups

TL;DR

This paper investigates whether certain linear rank inequalities, valid for vector spaces, also hold for finite groups, finding that groups of order pq satisfy them, while some larger groups like S4 violate some inequalities.

Contribution

It proves that groups of order pq satisfy ten specific group inequalities and identifies the smallest group, S4, that violates some of these inequalities.

Findings

Groups of order pq satisfy all ten inequalities.

S4 is the smallest group violating some inequalities.

Partial results for groups of order p^2q.

Abstract

We consider ten linear rank inequalities, which always hold for ranks of vector subspaces, and look at them as group inequalities. We prove that groups of order pq, for p,q two distinct primes, always satisfy these ten group inequalities. We give partial results for groups of order $p^2q$, and find that the symmetric group $S_4$ is the smallest group that yield violations, for two among the ten group inequalities.