The Automorphism Group of a Finite p-Group is Almost Always a p-Group

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-length, the number of generators, and p. The proof of this theorem depends on a variety of topics: counting subgroups of a p-group; analyzing the lower p-series of a free group via its connection with the free Lie algebra; counting submodules of a module via Hall polynomials; and using numerical estimates on Gaussian coefficients.Comment: 38 pages, to appear in the Journal of Algebra; improved references, changes in terminolog

Normal subgroups in the Cremona group (long version)

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.Comment: With an appendix by Yves de Cornulier. Numerous but minors corrections were made, regarding proofs, references and terminology. This long version contains detailled proofs of several technical lemmas about hyperbolic space

"The Great Event of the Fortnight”: Steamship Rhythms and Colonial Communication

This paper engages with Tim Cresswell’s ‘contellations of mobility’ in order to contribute some understanding of historical maritime rhythms. The empirical focus is upon a steamship mail service in the post-emancipation Caribbean. In examining this communications network, it is stressed that while those managing the network valorised predictable efficiency, ‘friction’ was prized by mercantile groups at the steamers’ ports of call. Thus, the different aspects of mobility signified differently across the network, and this historical case study reinforces the resonance of slowness and stoppage time. The synchronisation of steamship arrivals with sociocultural norms in the Caribbean colonies also necessitated the adaptation of mail service rhythms. Through a focus on shipping operations, this paper proposes to temper our understanding of the role of steamship technology in empire. The influence of colonies on the metropole encompassed an alteration of the rhythms of imperial circulation, and it is within the maritime arena that these realities came into sharp focus

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn2/log2n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate

A Counterexample Regarding Labelled Well-Quasi-Ordering

Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns