Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Thakur, Maneesh (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

The main theme of this thesis is the study of exceptional algebraic groups via their subgroups. This theme has been widely explored by various authors (Martin Leibeck, Gary Seitz, Adam Thomas, Donna Testerman to mention a few), mainly for split groups ([26], [27], [28], [60] ). When the field of definition k of the concerned algebraic groups is not algebraically closed, the classification of k-subgroups is largely an open problem. In the thesis, we mainly handle the cases of simple groups of type F4 and G2 defined over an arbitrary field. These may not be split over k. We first determine the possible simple k-subgroups of a fixed simple k-algebraic group of type G2 or F4 and then, find conditions for a simple k-algebraic group to embed in a given group of type G2 or F4.One knows that a group of type G2 over a field k arises as the group of automorphisms of an octonion algebra over k and similarly, groups of type F4 over k arise from Albert algebras. We exploit the structure of these algebras to derive our results. On the way we also obtain some results on these algebras, which may be of independent interest. For example, we derive a group theoretic characterization of first Tits construction Albert algebras (Theorem 10.2.3). We also prove a group theoretic characterization of Albert algebras A with f5(A) = 0 (Theorem 9.1.2). Other than these results, we prove some results on generation of the groups discussed above by their simple k-subgroups and k-tori, determining the number of such subgroups required in each case. The results in this thesis have been partly published in ([10]) and partly under submission ([9]).We now sketch below an outline of the work done in this thesis, introducing some notation on the way, which will be necessary in the Main results section. Let K be an algebraically closed field. The classification of semisimple algebraic groups over K is well understood.Theorem 0.0.1 (Chevalley Classification Theorem) Two semisimple linear algebraicgroups are isomorphic if and only if they have isomorphic root data. For each root datum there exists a semisimple algebraic group which realizes it. The simple algebraic groups have irreducible root systems or equivalently, have connected Dynkin diagrams. Irreducible root systems fall into nine types, called the CartanKilling types, labelled as An, Bn, Cn, Dn, E6, E7, F4, G2. The first four types exisit for each natural number n, while the remaining five types are just one in each case. Simple groups with root system or Dynkin diagram of types An, Bn, Cn, Dn are called classical groups and the simple groups with root systems of type E6, E7, E8, F4, G2 are called exceptional groups. Let G be a simple algebraic group over a field k. By the type of G we mean the Cartan-Killing type of the root system of the group G ⊗ k, obtained by extending scalars to an algebraic closure k of k.Let G be a simple linear algebraic group over K. Then corresponding to any subdiagram of the Dynkin diagram of G, there exists a subgroup of G which realizes it, i.e. has the subdiagram as its Dynkin diagram. But this fails to hold for a non-algebraically closed field. For example, over a non-algebraically closed field k a connected simple algebraic group G may not have any subgroup of type A1, though the Dynkin diagram of G always has A1 as a subdiagram (see Remark 10.2.2). Hence over a non-algebraically closed field k, it is important to know what are all simple k-subgroups of G. In the thesis we answer this for groups of type A2, G2 and F4. We prove that when G is a k-group of type F4 (resp. G2) arising from an Albert (resp. octonion) division algebra then the possible type of a simple k-subgroup of G is A2 or D4 (resp. A1 or A2). The knowledge of these simple k-subgroups is a useful tool in studying these groups. This motivates the Problem : Find conditions under which a given simple k-group of type A1 or A2 embeds over k in a simple k-group of type G2 or F4.In the thesis we study conditions which control the k-embeddings of simple algebraic groups of type A1 and A2 in simple groups of type G2 and F4 as well as k-embeddings of rank-2 k-tori in simple groups of type A2, G2 and F4. This is done via the mod-2 invariants attached to these groups.


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