The cyclic structure of maximal tori of the finite classical groups

Non-Abelian composition factors of m-rational groups

“…None of the work contained in this section is our original work, and we rely heavily on the work provided in [5], [50], and [27].…”

“…Then, for each 3 ≤ n ≤ 22, we may choose q 0 as follows: Verifying which of these (finitely many) pairs (n, q) satisfy φ(N ) n | 2| Out(S)| (see the Appendix for the complete set of computations), we determine that S = PSL n (q) can occur as a composition factor of a quadratic rational group only if (n, q) ∈ {(2, 4), (2,5), (2, 7), (2,8), (2,9), (2,11), (2,16), (2,19), (2,23), (2,27), (2,31), (3,2), (3,3), (3,4), (3,7), (3,16) is to occur as a composition factor of a quadratic rational group, it must be the case…”

Spectra of finite symplectic and orthogonal groups

“…Group Ω 2n+1 (q) is simple if pair (n, q) is distinct from (1, 2), (1, 3), (2,2). Group Ω ε 2n (q) has the center of order (4, q n − ε1)/2 and its factor group by the center is denoted by P Ω ε 2n (q).…”

“…It is well known that each semisimple element of a group of Lie type is contained in some maximal torus of this group. In [2] a description of cyclic structure of maximal tori in all groups under consideration was obtained, and thus the semisimple parts of spectra of these groups was described. Hence, it remains to describe the composite parts of the spectra.…”

“…Observe that groups H ξ 1j and H ζ 1j consist of diagonal matrices. Moreover, a group of diagonal matrices with blocks A j and B j along the diagonal, where A j ∈ H ξ 1j and B j ∈ H ζ 1j , in contained in Sp 2n 1 (F p ) and conjugate in it to a maximal torus of Sp 2n 1 (q), corresponding to the pair of partitions ξ (1) and ζ (1) (see [2,Proposition 3.1]). Thus, we proved…”