Group Theory in Trento 2012

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Abstracts

Some Nottingham algebras
(with S. Mattarei)
We investigate a class of infinite-dimensional modular Lie algebras, graded over the positive integers and satisfying a certain narrowness condition, modeled on an analogous condition for pro-\(p\) groups. We identify these Lie algebras with loop algebras of certain simple Lie algebras of Hamiltonian type, suitable graded. We prove some properties of the Laguerre polynomial of a derivation which play a role in the construction of the suitable grading of the Hamiltonian algebras.

An effective version of the Lazard correspondence
The Lazard correspondence defines a bijection between \(p\)-groups of order \(p^n\) and nilpotency class \(< p\), and Lie rings of the same order and nilpotency class. We show how we can effectively set up this correspondence for a given \(p\)-group. This involves computing the inverse BCH-formulas up to a given weight. We discuss an application of the Lazard correspondence involving non-commuting graphs. This is joint work with Willem de Graaf and Michael Vaughan-Lee.

Character Degrees of Finite \(p\)-Groups by Coclass
Let \((G_k \mid k \in \mathbb{N}_0)\) be a coclass family of finite \(p\)-groups and let \(l\) be a nonnegative integer. We show that there is a polynomial \(f_l \in \mathbb{Q}[X]\) with \(\deg(f_l) \leq d\), where \(d\) is the dimension of the associated pro-\(p\)-group, such that \(f_l(p^k) = \#\{ \chi \in \text{Irr}(G_k) \mid \chi(1) = p^l \}\) holds for every large enough \(k\).

Can we classify \(p\)-groups up to isomorphism?
I shall consider the issues involved in trying, in some significant sense, to classify the class of all \(p\)-groups up to isomorphism, with particular reference to the coclass project.

On the minimal permutation degrees of abelian quotients of finite \(p\)-groups
For a finite group \(G\), we denote by \(\mu(G)\) the minimal permutation degree of \(G\), that is the minimum degree of a faithful permutation representation of \(G\). Clearly, for any subgroup \(H\) of \(G\), \(\mu(H)\leq \mu(G)\). The same does not hold in general for quotient groups \(G/N\). In this talk we present some results on relations between \(\mu(G)\) and \(\mu(G/G')\), when \(G\) is a finite \(p\)-group. We also discuss their connection with a conjecture by Easdown and Praeger on the minimal degree of abelian quotients of finite groups

Profinite groups of finite virtual length
In an (infinite) profinite group \(G\) every series can be properly refined so \(G\) cannot have a composition series. We introduce then a notion of virtual composition series in a profinite group, namely a series such that all its refinements have the same number of sections of infinite order. If such a series exists, we say that \(G\) has finite virtual length and we develop an analogue of Jordan-Hölder theory in this context, with the hereditarily just-infinite pro-\(p\) groups playing the role of finite simple groups.

Unipotent automorphisms of residually finite groups
(with C. Casolo)
Let \(G\) be any group and \(x\) an automorphism of \(G\). The automorphism \(x\) is said to be unipotent if there exists \(n\) such that \([g,_n x] = 1\) for all \(g \in G\). Commutators are taken in the holomorph of \(G\) and, following the usual notation, \([g,_n x]\) is defined inductively by \([g,_0 x] = g\) and, when \(n>0\), \([g,_n x] = [[g,_{n-1} x], x]\). A unipotent automorphism \(x\) could also be seen as a left \(n\)-Engel element in the group \(G\langle x\rangle\). When \(G\) is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this talk we shall focus on groups of unipotent automorphisms of residually finite groups, proving that such groups are locally nilpotent. It is easily seen that, instead of residually finite groups, one can consider profinite groups, and thus take advantage of the well-developed theory available for groups in this class. The crucial step of the proof is to show that the result holds when \(G\) is a finitely generated pro-\(p\)-group. Once this fact has been established, the general case follows using some recent work of Crosby and Traustason on the structure of Engel groups.

Finite \(p\)-groups that determine \(p\)-nilpotency locally
Let \(G\) be a finite group, and let \(P\) be a Sylow \(p\)-subgroup of \(G\). It is possible that \(N_G(P)\) is \(p\)-nilpotent, but \(G\) is not \(p\)-nilpotent. However, this is not possible if \(P\) is abelian. We say that a finite \(p\)-group \(P\) determines \(p\)-nilpotency locally, if the following holds: For any finite group \(G\) with Sylow \(p\)-subgroups isomorphic to \(P\) and \(N_G(P)\) \(p\)-nilpotent it follows that \(G\) is also \(p\)-nilpotent.
Let \(Y_1\simeq C_p\) be the Sylow \(p\)-subgroup of \(S_p=\textrm{Sym}(p)\), and let \(Y_k\) be the pull-back of the diagram \[\begin{array}{ccc} C_{p^k} & \longrightarrow & C_p\\ \uparrow & & \uparrow\\ Y_k & \longrightarrow & Y_1 \end{array} \] Using results from J. G. Thompson and G. Glauberman we will show that any finite \(p\)-group \(P\), \(p\) odd, with no subgroups isomorphic to \(Y_k\), \(k\geq 1\), determines \(p\)-nilpotency locally. This theorem generalizes a result of T. Yoshida who proved a similar statement for \(Y_1\)-free finite \(p\)-groups.

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