Graham Higman's PORC theorem

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments

Graded Lie algebras of maximal class IV

We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of odd characteristic.Comment: 38 pages. See also http://www-math.science.unitn.it/~caranti/ and http://users.ox.ac.uk/~vlee

Non-PORC behaviour of a class of descendant pp-groups

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture

A softer look at MCG--6-30-15 with XMM-Newton

We present analysis and results from the Reflection Grating Spectrometer during the 320 ks XMM observation of the Seyfert 1 galaxy MCG-6-30-15. The spectrum is marked by a sharp drop in flux at 0.7 keV which has been interpreted by Branduardi-Raymont et al. as the blue wing of a relativistic OVIII emission line and by Lee at al. as a dusty warm absorber. We find that the drop is well explained by the FeI L2,3 absorption edges and obtain reasonable fits over the 0.32-1.7 keV band using a multizone, dusty warm absorber model. Some residuals remain which could be due to emission from a relativistic disc, but at a much weaker level than from any model relying on relativistic emission lines alone. A model based on such emission lines can be made to fit if sufficient (warm) absorption is added, although the line strengths exceed those expected. The EPIC pn difference spectrum between the highest and lowest flux states of the source indicates that this is a power-law in the 3-10 keV band which, if extrapolated to lower energies, reveals the absorption function acting on the intrinsic spectrum, provided that any emission lines do not scale exactly with the continuum. We find that this function matches our dusty warm absorber model well. The soft X-ray spectrum is therefore dominated by absorption structures, with the equivalent width of any individual emission lines in the residuals being below about 30 eV. (abridged)Comment: 8 pages, 10 figures, submitted to MNRA

A long hard look at MCG-6-30-15 with XMM-Newton and BeppoSAX

We summarise the primary results from a 320 ks observation of the bright Seyfert 1 galaxy MCG-6-30-15 with XMM-Newton and Beppo-SAX.Comment: 4 pages, 6 figures. Proc. of the meeting: "The Restless High-Energy Universe" (Amsterdam, The Netherlands), E.P.J. van den Heuvel, J.J.M. in 't Zand, and R.A.M.J. Wijers Ed

5-Engel Lie algebras

We prove that 5-Engel Lie algebras over a field of characteristic zero, or over a field of prime characteristic $p>7$, are nilpotent of class at most 11. We also prove that if $G$ is a finite 5-Engel $p$-group for $p>7$ then $G$ is nilpotent of class at most 10.Comment: 13 pages, some typos correcte