The Chevalley group G2(2) of order 12096 and the octonionic root system of E7

Mehmet Koca, Ramazan Koç*, Nazife Ö Koca

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 root system. The other three maximal subgroups of orders 432; 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E6 × U(1), SU(2) × SO(12) and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

Original languageEnglish
Pages (from-to)808-823
Number of pages16
JournalLinear Algebra and Its Applications
Issue number2-3
Publication statusPublished - Apr 15 2007


  • Group structure
  • M-Theory
  • Quaternions
  • Subgroup structure

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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