Intersection graphs of almost subnormal subgroups in general skew linear groups

Xuan Hai Bui (1) , Binh-Minh Bui-Xuan (2) , Le Van Chua (3) , Mai Hoang Bien (1)

1 HCMUS - Ho Chi Minh City University of Science
2 APR - Algorithmes, Programmes et Résolution
3 VNU-HCM - Vietnam National University, Ho Chi Minh City / Đại học Quốc gia TP. Hồ Chí Minh

Let D be a division ring, n a positive integer, and GL_n(D) the general linear group of degree n over D. In this paper, we study the induced subgraph of the intersection graph of GL_n(D) generated by all non-trivial proper almost subnormal subgroups of GL_n(D). We show that this subgraph is complete if it is non-null. This property will be used to study subgroup structure of a division ring. In particular, we prove that every non-central almost subnormal subgroup of the multiplicative group D* of a division ring D contains a non-central subnormal subgroup of D*.

Domaines: Anneaux et algèbres [math.RA]; Mathématique discrète [cs.DM]

Xuan Hai Bui, Binh-Minh Bui-Xuan, Le Van Chua, Mai Hoang Bien. Intersection graphs of almost subnormal subgroups in general skew linear groups. 2020. ⟨hal-02997959⟩