Research

Abstract. We introduce an abstract notion of a 3D-rotation module for a group $G$ that does not require the module to carry a vector space structure, a priori nor a posteriori. Our first result shows that, under an expected irreducibility-like assumption, the only finite $G$ with such a module are those already known from the classical setting: $\operatorname{Alt}(4)$, $\operatorname{Sym}(4)$, and $\operatorname{Alt}(5)$. Our second result then studies the module structure when $G = \operatorname{Alt}(5)$ and shows that, under certain natural restrictions, it is fully determined and generalizes that of the classical icosahedral module.

We include an application to the recently introduced setting of modules with an additive dimension, a general setting allowing for simultaneous treatment of classical representation theory of finite groups as well as representations within various well-behaved model-theoretic settings such as the $o$-minimal and finite Morley rank ones. Leveraging our recognition result for icosahedral modules, we classify the faithful $\operatorname{Alt}(5)$-modules with additive dimension that are dim-connected of dimension $3$ and without $2$-torsion.

Abstract. We show that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for example, when the stabilizer is solvable or when $r\le 5$. This makes progress on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive permutation group of finite Morley rank on a set of rank $r$ is of the form $\operatorname{PGL}_{r+1}(F)$ acting naturally on $\mathbb{P}^r(F)$.

Our proof is assembled from three key ingredients that are independent of the main theorem -- these address actions of $\operatorname{Alt}(n)$ on $L$-groups of finite Morley rank, generically $2$-transitive actions with abelian point stabilizers, and simple groups of rank $6$.

Abstract. Working in the general context of "modules with an additive dimension," we complete the determination of the minimal dimension of a faithful Alt(n)-module and classify those modules in three of the exceptional cases: 2-dimensional Alt(5)-modules in characteristic 2, 3-dimensional Alt(5)-modules in characteristic 5, and 3-dimensional Alt(6)-modules in characteristic 3. We also highlight the remaining work needed to complete the classification of the faithful Alt(n)-modules of minimal dimension for all n; these open problems seem well suited as projects for advanced undergraduate or master's students.

Abstract. We investigate faithful representations of Alt(n) as automorphisms of a connected group $G$ of finite Morley rank. We target a lower bound of $n$ on the rank of such a nonsolvable $G$, and our main result achieves this in the case when $G$ is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable $G$ by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank.

Abstract. We revisit, clarify, and generalise classical results of Dickson and (much later) Wagner on minimal Sym(n)- and Alt(n)-modules. We present a new, natural notion of 'modules with an additive dimension' covering at once the classical, finitary case as well as modules definable in an o-minimal or finite Morley rank setting; in this context, we fully identify the faithful Sym(n)- and Alt(n)-modules of least dimension.

Abstract. We revisit the geometry of involutions in groups of finite Morley rank. The focus is on specific configurations where, as in $\operatorname{PGL}_2(\mathbb{K})$, the group has a subgroup whose conjugates generically cover the group and intersect trivially. Our main result is the subtle yet strong statement that in such configurations the conjugates of the subgroup may not cover all strongly real elements. As an application, we unify and generalise numerous results, both old and recent, which have exploited a similar method; though in fact we prove much more. We also conjecture that this path leads to a new identification theorem for $\operatorname{PGL}_2(\mathbb{K})$, possibly beyond the finite Morley rank context.

Abstract. In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank $(X,G)$ is at most $n+2$ where $n$ is the Morley rank of $X$. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by $\operatorname{PGL}_{n+1}(\mathbb{F})$ acting naturally on projective $n$-space. We solve the problem under the two additional hypotheses that (1) $(X,G)$ is $2$-transitive, and (2) $(X-\{x\},G_x)$ has a definable quotient equivalent to $(\mathbb{P}^{n-1}(\mathbb{F}),\operatorname{PGL}_{n}(\mathbb{F}))$. The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.

Abstract. We show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$.

Abstract. We show that the only transitive and generically $4$-transitive action of a group of finite Morley rank on a set of Morley rank $2$ is the natural action of $\operatorname{PGL}_3$ on the projective plane.

Abstract. A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.

Abstract. We show that any simple group of Morley rank $4$ must be a bad group with no proper definable subgroups of rank larger than $1$. We also give an application to groups acting on sets of Morley rank $2$.

Abstract. We show that for a wide class of groups of finite Morley rank the presence of a split $BN$-pair of Tits rank 1 forces the group to be of the form $\operatorname{PSL}_2$ and the $BN$-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian $2$-subgroups and show that all such Moufang sets are standard (and thus associated to $\operatorname{PSL}_2(F)$ for $F$ an algebraically closed field of characteristic not $2$) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are $L$-groups and the root groups are not simple.

Abstract. We study groups of finite Morley rank with a split $BN$-pair of Tits rank $1$ in the case where the normal complement to $B\cap N$ in $B$ is infinite and abelian. For such groups, we give conditions ensuring that the standard action of $G$ on the cosets of $B$ is isomorphic to the natural action of $\operatorname{PSL}_2(F)$ on $\operatorname{P}_1(F)$ for $F$ an algebraically closed field. In particular, we show that $\operatorname{SL}_2(F)$ and $\operatorname{PSL}_2(F)$ are the only infinite quasisimple $L^*$-groups of finite Morley rank possessing a split $BN$-pair of Tits rank $1$ where the normal complement to $B\cap N$ in $B$ is infinite abelian. Our approach is through the theory of Moufang sets and is tied to work attempting to classify the abelian Moufang sets of finite Morley rank.

Abstract. In this paper we study special Moufang sets, $\mathbb{M}(U,\tau)$, with $U$ abelian under the additional restriction that they have finite Morley rank. Our result states that the little projective group of such a Moufang set must be isomorphic to $\operatorname{PSL}_2(K)$ for $K$ an algebraically closed field provided that $U$ has characteristic two and that infinitely many endomorphisms of $U$ centralize the Hua subgroup. This complements a result of De Medts and Tent that addresses the characteristic not two case.

More info. These notes were created to support a second course in undergraduate abstract algebra. Beginning from fields (and the complex numbers), the notes work their through field extensions (generally), rings, algebraic field extensions, an introduction to Galois theory, and, ultimately, insolvability of the general quintic.

More info. The code contains a handful of functions for computing the height and relational complexity of a finite permutation group. As an example, RCompDataLatex.pdf - contains formatted relational complexity data for most of the primitive permutation groups of degree at most 100.

More info. These notes were created to support a four-day mini-course on groups of finite Morley rank given at the Hausdorff Institute for Mathematics in November 2018 during the Trimester Program on Logic and Algorithms in Group Theory.

More info. This is a short (approxiamtely two-page) section to explore modular arithmetic (IBL style) from the point of view of equivalence relations.