# Quasi-split group

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{{Short description|Linear algebraic group}}
In mathematics, a '''quasi-split group''' over a [field](/source/field_(mathematics)) is a [reductive group](/source/reductive_group) with a [Borel subgroup](/source/Borel_subgroup) defined over the field.  [Simply connected](/source/Simply_connected) quasi-split groups over a field correspond to actions of the absolute [Galois group](/source/Galois_group) on a [Dynkin diagram](/source/Dynkin_diagram).

==Examples==
All [split group](/source/split_group)s (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.

{{harvtxt|Lang|1956}} showed that all simple algebraic groups over finite fields are quasi-split.

Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups ''O''<sub>''n'',''n''+2</sub>, the unitary groups ''SU''<sub>''n'',''n''</sub> and ''SU''<sub>''n'',''n''+1</sub>, and the form of ''E''<sub>6</sub> with signature 2.

==References==
*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebraic groups over finite fields | jstor=2372673 |mr=0086367 | year=1956 | journal=[American Journal of Mathematics](/source/American_Journal_of_Mathematics) | issn=0002-9327 | volume=78 | pages=555–563 | doi=10.2307/2372673}}

Category:Linear algebraic groups

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