Group inequalities

Abstract

The Ingleton inequality is a significant concept in both information theory and matroid theory, providing a crucial link between event probabilities and the im- portance of sets within matroids. This final year thesis explores the application of group theory to investigate the validity of this inequality in previously unexplored scenarios. Beginning with an examination of the symmetric group S5, known to violate the inequality, we break down the known proof for abelian groups to in- vestigate the reasons. We then extend our study to include the family of dihedral groups, by examining the structure of dihedral groups, and its implication on the Ingleton inequality. Our investigation concludes with the application of Goursat’s Lemma to analyze how the inequality behaves on the direct product of abelian and non-abelian groups. Through these methodologies, the aim is to deepen the understanding of the Ingleton inequality and uncover potential insights with impli- cations for both theoretical and practical domains within information and matroid theory.