Researcher Đậu Thị Huế with supervisors and the examination committee after the successful defence.

Researcher Đậu Thị Huế (class 2018) has successfully defended a doctoral thesis at the university level in the field of Algebra and Number Theory at VNUHCM-University of Science. The dissertation, entitled “Symmetric Sets of Subgroups in Division Rings with Involution”, was supervised by Prof. Bùi Xuân Hải, former Vice Dean of the Faculty of Mathematics and Computer Science.

This thesis represents an in-depth contribution to the field of pure mathematics, specifically in ring theory and group theory – two foundational branches of modern algebra. The research investigates “symmetric sets” within specialised algebraic structures known as division rings with involution.

These structures can be understood as mathematical systems that possess not only the basic operations of addition and multiplication, but also a distinctive “reflection” operation – known as involution – that introduces internal symmetry into the structure.

The thesis offers new insights into the classification and internal structure of algebraic systems with involution. A noteworthy result is the generalisation of a classical theorem concerning the decomposition of algebraic structures, providing conditions under which a subgroup can reflect the global structure of the ring.

In particular, the thesis clarifies the criteria under which an algebraic system can be described by a quaternion ring – a four-dimensional algebra over a base field, widely applied in theoretical physics, computer graphics, and quantum mechanics. [*]

Another significant contribution is the establishment of a relationship between the symmetry of subgroups and the global structure of the ring. This holistic approach demonstrates that satisfying certain specific subgroup-level conditions may lead to defining characteristics of the entire system.

The research outcomes not only advance the theory of involution and division rings, but also open new directions for further investigation in abstract algebra. In the concluding chapter, the thesis proposes future research pathways, such as exploring other types of involution (e.g. second-type involution) and extending the analysis to less-studied classes of rings beyond the classical framework.

The successful defence marks a significant milestone in the academic journey of researcher Đậu Thị Huế at VNUHCM-University of Science. Warmest congratulations on this achievement.

[*] Note: Applications of quaternion rings in physics, computer graphics, and related fields are general mathematical applications and are not direct contents of this dissertation.

_{▪︎ Detailed information on the dissertation has been published on the official doctoral thesis portal of VNUHCM-University of Science.}

PMN_Photo credit: Prof. Bùi Xuân Hải