View source for Quasi-split group

''quasi-splittable group, over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767101.png" />''

An affine algebraic group (cf. [[Affine group|Affine group]]; [[Algebraic group|Algebraic group]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767102.png" /> containing a [[Borel subgroup|Borel subgroup]] defined over the same field. Every affine algebraic group becomes a quasi-split group for some extension of the ground field, for example, over the algebraic closure of this field. Every affine algebraic group defined over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767103.png" /> is quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767104.png" />. See also [[Split group|Split group]].



====Comments====
A group can be quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767105.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767107.png" />-quasi-split, without being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767108.png" />-split.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,   "Linear algebraic groups" , Springer  (1975)  pp. Sect. 35.1</TD></TR></table>