Automorphism groups of the pure integral octonions and applications in physics

VH ABSTRACT The automorphism groups of the pure integer octonions are studied. The seven-dimensional matrix representation of the automorphism group of pure integer octonions constituting the root system of E7 is constructed. It is shown that G2(2) which is a finite subgroup of the exceptional Lie group G2, of order 12096, has four maximal subgroups. The structure of G2(2) and its maximal subgroups are identified and their possible implications in physics are discussed. The maximal subgroups of G2(2) having the orders 432,192,192 and 336 preserve the octonionic root systems of E6, SO(12), SU(2)3xSO(8) and SU(8), respectively. Matrix representation of the automorphism group of the octonionic imaginary units ±ei (i=1,...7) is also constructed. The group 23-PSL2(7) of order 1344 is the non- split extension of the elementary Abelian group of order 8 by Klein's simple group PSL2(7) of order 168. The maximal subgroups of 23-PSL2(7) and their structure are analysed. Seven-dimensional matrix representation of another group 23:PSL2(7) of order 1344 which is the split extension of the elementary Abelian group of order 8 by PSL2(7) is constructed. Its maximal subgroups are studied. The character tables and the conjugacy classes of the groups G2(2), 23-PSL2(7), 23:PSL2(7) and all maximal and Sylow subgroups are computed.