Thomas Meixner

Flag-transitive extensions ofC n geometries

We consider locally Cn-geometries where all planes are circular spaces (i.e., complete graphs). We call them extensions of Cn-geometries or (c.Cn) geometries, for short. We give a classification of finite flag-transitive (c.Cn) geometries) geometries when n≥3. A classification is given also in the case of n=2 under the hypothesis that residues of points are thick classical generalized quadrangles.

The 2-adic affine building of type Ã2 and its finite projections

A certain “free” group U is constructed that is generated by three elements of order 3 which pairwise generate a Frobenius group of order 21 and it is shown that U operates regularly on the affine building of type A2 over the field of 2-adic numbers. As a result an infinite series of finite rank 3 geometries is obtained whose rank 2 residues are projective planes of order 2, and which possess a regular automorphism group isomorphic to SL3(p) or SU3(p) for some prime p.

The fitting length of solvableHpn-groups

For a (finite) groupG and some prime powerpn, theHpn-subgroupHpn (G) is defined byHpn (G)=〈xεG|xpn≠1〉. A groupH≠1 is called aHpn-group, if there is a finite groupG such thatH is isomorphic toHpn (G) andHpn (G)≠G. It is known that the Fitting length of a solvableHpn-group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableHpn-groups having Fitting lengthn, this result is “almost” best possible.

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp2 possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.

Locally finite classical tits chamber systems with transitive group of automorphisms in characteristic 3

The classification of locally finite classical Tits chamber systems C of finite rank admitting a transitive group G of automorphisms, such that the stabilizer in G of some chamber is finite, is now complete by work of several authors. In the following, the case, that on a rank 2 residue of C some exceptional flag-transitive subgroup of Aut(U4(3)) or Aut(PSp4(3)) is induced by G, is treated.

Solvable Groups Admitting an Automorphism of Prime Power Order Whose Centralizer Is Small

If a finite solvable group G admits an automorphism x of order p”, p some prime, such that x has no nontrivial fixed points on G, then the Fitting length of G is at most n. This result goes back to Hoffman [8], Shult [lo] and Gross [4,5], although in [8] and [lo] it is only proved under some additional hypothesis to the prime p and the primes dividing the order of G, and Gross could only show that the Fitting length is at most 2n - 2 for p = 2. The complete proof is contained, however, in [2], and Berger treats a much more general situation: the automorphism groups he allows do not have to be cyclic of prime power order, but only nilpotent. But let us still consider the automorphism x of order p”. We see, that the fixed point free action of x on G forces the structure of G to be “not too wild” in terms of the order of x. Now it is clear, that if x is allowed to have m fixed points on G, and (IGI, p) = 1, then also the Fitting length of G is bounded by some function of n and m, since the sections F,, ,,( G)/Fi- ,(G) must contain fixed points of x by the above theorem. But it seems more reasonable to look, whether G might still be “close to” having Fitting length at most n, differing from that only by some function of m. This was done in [6] and [7] for x of order p, i.e., it was shown that there is some function f such that (G: F(G)( is bounded by f (p, m), if x has order p, coprime to ICI, and lC,(x)l

A family of multiply extended grids

We construct an infinite family{Γn}n=5 of finite connected graphsΓn that are multiple extensions of the well-known “extended grid” discovered in [1] (which is isomorphic toΓ5). The graphsΓn are locallyΓn−1 forn > 5, and have the following property: the automorphism groupG(n) ofΓn permutes transitively the maximal cliques ofΓn (which aren-cliques) and the stabilizer of somen-cliqueπ ofΓn inG(n) inducesΣn on the vertices ofπ. Furthermore we show that the clique complexes of the graphsΓn are simply connected.

Folding Down Classical Tits Chamber Systems

Let Ĝ be some classical p-adic group, then the existence of a discrete subgroup of G acting chamber transitively on the affine building Δ of Ĝ is a rare occurency, if the rank of the building is at least three. This is part of a theorem by Kantor, Liebler and Tits. In the cases where such a subgroup exists, one has always p = 2 or 3 and constructions of the exceptional groups can be found in [4], [5], [6], [7], [8] and [12]. Many (if not all) of the constructions — the starting point being [4] — made use of some “diagram automorphism” acting on the vertices contained in a chamber of Δ. These automorphisms of the resulting groups can, however, also be used to fold down the groups — to get subgroups acting on buildings of smaller rank “contained in Δ”; again this idea was first shown to be successful in ([4], section 6). In the following we apply this method to some rank-3-cases; thereby we get groups acting on rank-2-buildings of affine type, i.e. on trees. As a result, we obtain small-dimensional faithful representations for the amalgamated sum of the maximal parabolics of some rank-2-Chevalley groups and for the two biggest groups in Goldschmidt’s list of all primitive amalgams of index (3.3)([3]).