On a conjecture of Gluck

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.Comment: 16 page

Gluck twist on a certain family of 2-knots

We show that by performing the Gluck twist along the 2-knot $K^2_{pq}$ derived from two ribbon presentations of the ribbon 1-knot $K(p,q)$ we get the standard 4-sphere $S^4$. In the proof we apply Kirby calculus.Comment: 11 pages, 12 figure

On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture

A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of verticesof $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels.The distinguishing number $D(G)$ of $G$ is then the smallest $r$ for which $G$ admits a distinguishing $r$-labeling.From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups,{\em Can. J. Math.} 35(1) (1983), 59--67),it follows that $D(T)=2$ for every cyclic tournament~$T$ of (odd) order $2q+1\ge 3$.Let $V(T)=\{0,\dots,2q\}$ for every such tournament.Albertson and Collins conjectured in 1999that the canonical 2-labeling $\lambda^*$ given by$\lambda^*(i)=1$ if and only if $i\le q$ is distinguishing.We prove that whenever one of the subtournaments of $T$ induced by vertices $\{0,\dots,q\}$or $\{q+1,\dots,2q\}$ is rigid, $T$ satisfies Albertson-Collins Conjecture.Using this property, we prove that several classes of cyclic tournaments satisfy Albertson-Collins Conjecture.Moreover, we also prove that every Paley tournament satisfies Albertson-Collins Conjecture

The ghost of Alcestis

This chapter considers a complex of materials centred on the Alcestis of Euripides and its reception history as an opera (Lully, Gluck) in early modern France. The interest of this particular text is that its operatic setting by Lully generated a polemic in the 1670s which initiated the ‘Querelle des anciens et des modernes’, a founding moment of modernity. This study thus interrogates the very notion of ‘ancient’ and ‘modern’ as it is deployed around the Alcestis, that is, as Lully and Gluck translated modernity into music. The fundamental question of this modernity would be ‘Who or what was returned to Admetus?’ My argument is that the ghost of Alcestis, written out, written over, or perhaps repressed the in 1674 Lully opera, re-emerges in the theoretical discourse of Racine and Perrault surrounding the opera as the general question of what, exactly, can be retrieved from antiquity. Equally, her ghostliness eliminated from the plot of Gluck’s 1776 Paris reform opera, it is her voice, her very music, which is invaded by the musical figure of the ghost. The theoretical frame for this study is formed by the notion of ‘hauntology’, a trend in recent critical and psychoanalytical work that attempts to link the theme of the ghost to textuality in general