A pseudo-reductive group is a smooth connected affine algebraic group over a field k which does not contain any nontrivial smooth connected normal unipotent subgroups defined over k. Such groups arise naturally as the quotient of any smooth connected affine algebraic k-group by the maximal smooth connected normal unipotent subgroup defined over k. For study of general affine algebraic groups it is important to know the structure and classification of pseudo-reductive groups. In a joint work with Brian Conrad and Ofer Gabber we have determined the structure and classification of these groups. In my talk I will explain the classification, and also mention group theoretic and arithmetic applications.
