Editing G2 (mathematics) (section)

{{Short description|Simple Lie group; the automorphism group of the octonions}}
{{DISPLAYTITLE:G<sub>2</sub> (mathematics)}}
{{Group theory sidebar |Topological}}
{{Lie groups |Simple}}

In [[mathematics]], '''G<sub>2</sub>''' is three simple [[Lie group]]s (a complex form, a compact real form and a split real form), their [[Lie algebra]]s <math>\mathfrak{g}_2,</math> as well as some [[algebraic group]]s. They are the smallest of the five exceptional [[simple Lie group]]s. G<sub>2</sub> has rank 2 and dimension 14. It has two [[fundamental representation]]s, with dimension 7 and 14.

The compact form of G<sub>2</sub> can be described as the [[automorphism group]] of the [[Octonion|octonion algebra]] or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional [[Real representation|real]] [[spinor]] [[Group representation|representation]] (a [[spin representation]]).