Narrow Results

Representations of SO(4, 2) are constructed using 4×4 and 2×2 matrices with elements in ℍ' ⊗ ℂ and the known isomorphism between the conformal group and SO(4, 2) is written explicitly in terms of the 4×4 representation. The Clifford algebra structure of SO(4, 2) is briefly discussed in this language, as is its relationship to other groups of physical interest.

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. We study element orders in proper covers of a finite simple group L of Lie type and prove that such a cover always contains an element whose order differs from the element orders of L provided that L is not L₄(q), U₃(q), U₄(q), U₅(2), or ³D₄(2).

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and $p$-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work, we handle all (simply connected) outer forms of $D_4$.

In this paper, we define a set of elementary orthogonal matrices similar to Vaserstein’s definition of elementary symplectic matrices. We prove that the elementary orthogonal group generated by these matrices is a conjugate of the Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal group defined on a free quadratic module with a hyperbolic summand.