The maximality of certain symplectic subgroups of unitary groups PSUn(K), n ≥ 4, (K any field admitting a non–trivial involutory automorphism) belonging to the class C5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated. Mathematics Subject Classification (2002): 20G40, 20G28

We apply Voronoi’s algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple Q-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.

Let [p] = {1, . . . , p} denote a totally ordered alphabet on p letters, and let α = (α1, . . . ,αm) ∈ [p1], β = (β1, . . . ,βm) ∈ [p2]. We say that α is order-isomorphic to β if for all 1≤ i< j ≤ m one has αi < α j if and only if βi < β j. For two permutations π ∈ Sn and τ ∈ Sk, an occurrence of τ in π is a subsequence 1 ≤ i1 < i2 < .. . < ik ≤ n such that (πi1 , . . . ,πik ) is order-isomorph...

THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ^ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then foll...

The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and th...