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We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures or extensions of previous conjectures, and prove a number of results. One of these is that, if $n\geq 21$, then there are at least eight irreducible characters of $S_n$, all of which have the same degree, and which have irreducible restriction to $A_n$. We explore similar questions about unipotent degrees of $\GL_n(q)$. We also make some remarks about how the experiments here shed light on posited algorithms for finding the largest irreducible character degree of $S_n$., Comment: 16 pages

Let $\mathcal{F}$ be a set of finite groups. A finite group $G$ is called an \emph{$\mathcal{F}$-cover} if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. An $\mathcal{F}$-cover is called \emph{minimal} if no proper subgroup of $G$ is an $\mathcal{F}$-cover, and \emph{minimum} if its order is smallest among all $\mathcal{F}$-covers. We prove several results about minimal and minimum $\mathcal{F}$-covers: for example, every minimal cover of a set of $p$-groups (for $p$ prime) is a $p$-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether $\{\mathbb{Z}_q,\mathbb{Z}_r\}$ has finitely many minimal covers, where $q$ and $r$ are distinct primes. Motivated by this, we say that $n$ is a \emph{Cauchy number} if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by $n$, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

This paper almost classifies the maximal subgroups of $E_7(q)$ for general $q$ a power of a prime $p$. Only four potential maximal subgroups are missing: $PSL_2(7)$ (unknown for $p\neq 2,3,7$), $PSL_2(8)$ ($p=2$), $PSL_2(9)=A_6$ ($p\neq 2,3$) and $PSL_2(13)$ ($p\neq 2,13$). In addition, there is one issue with the precise structure with the positive-dimensional maximal subgroup of type $A_2$. We are able to give a complete determination of the maximal subgroups for $E_7(4)$., Comment: 50 pages

We develop a theory of Ennola duality for subgroups of finite groups of Lie type, relating subgroups of twisted and untwisted groups of the same type. Roughly speaking, one finds that subgroups $H$ of $\mathrm{GU}_d(q)$ correspond to subgroups of $\mathrm{GL}_d(-q)$, where $-q$ is interpreted modulo $|H|$. Analogous results for types other than $\mathrm A$ are established, including for exceptional types where the maximal subgroups are known, although the result for type $\mathrm D$ is still conjectural. Let $M$ denote the Gram matrix of a non-zero orthogonal form for a real, irreducible representation of a finite group, and consider $\alpha=\sqrt{\det(M)}$. If the representation has twice odd dimension, we conjecture that $\alpha$ lies in some cyclotomic field. This does not hold for representations of dimension a multiple of $4$, with a specific example of the Janko group $\mathrm J_1$ in dimension $56$ given. (This tallies with Ennola duality for representations, where type $\mathrm D_{2n}$ has no Ennola duality with ${}^2\mathrm D_{2n}$.)

We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type $E_8$ in characteristic $3$. This has type $F_4$, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck--Seitz. We also prove a result about the finite group $H={}^3\!D_4(2)$, that if $H$ embeds in $E_8$ (in any characteristic $p$) and has two composition factors on the adjoint module then $p=3$ and $H$ lies in this new maximal $F_4$ subgroup., Comment: 12 pages. Proc. Amer. Math. Soc., to appear

This document is the first iteration of an attempt to collate information about small-rank groups of Lie type over small fields, and their representation theory over the defining field. This information is important in the author's work on subgroup structure of exceptional groups of Lie type. The most important information in that work is information about $\mathrm{Ext}^1$ between simple modules, and so in Version 1 of this document, that data is almost all of the data available. In addition, a lot of information about the dimensions of the simple and Weyl modules is included. More generally, one may expect to include details about the socle structure of the projective modules, Jordan block structure of the action of unipotent elements, decompositions of symmetric and exterior powers of simple modules, and tensor products of modules, traces of semisimple elements and so on. The ideal place for such information is a dedicated website, connected to a database that could be queried to produce the information required. This document, while imperfect, will have to suffice for now., Comment: 80 pages

This article produces a complete list of all maximal subgroups of the finite simple groups of type $F_4$, $E_6$ and twisted $E_6$ over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maximal subgroup of ${}^2\!F_4(8)$, correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the $27$-dimensional minimal module for $E_6$. We provide a collection of auxiliary Magma files that prove the author's computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text., Comment: 73 pages

We study embeddings of groups of Lie type $H$ in characteristic $p$ into exceptional algebraic groups $\mathbf G$ of the same characteristic. We exclude the case where $H$ is of type $\mathrm{PSL}_2$. A subgroup of $\mathbf G$ is \emph{Lie primitive} if it is not contained in any proper, positive-dimensional subgroup of $\mathbf G$. With a few possible exceptions, we prove that there are no Lie primitive subgroups $H$ in $\mathbf G$, with the conditions on $H$ and $\mathbf G$ given above. The exceptions are for $H$ one of $\mathrm{PSL}_3(3)$, $\mathrm{PSU}_3(3)$, $\mathrm{PSL}_3(4)$, $\mathrm{PSU}_3(4)$, $\mathrm{PSU}_3(8)$, $\mathrm{PSU}_4(2)$, $\mathrm{PSp}_4(2)'$ and ${}^2\!B_2(8)$, and $\mathbf G$ of type $E_8$. No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of $\mathbf G$. This has the consequence that, with the same exceptions, any almost simple group with socle $H$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$, $E_7$ and $E_8$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means., Comment: vi+214pp

We determine the group of endotrivial modules (as an abstract group) for $G$ a (quasi)simple group of sporadic type, extending previous results in the literature. In many sporadic cases we directly construct the subgroup of trivial-source endotrivial modules. We also resolve the question of whether certain simple modules for sporadic groups are endotrivial, posed by Lassueur, Malle and Schulte, in the majority of open cases. The results rely heavily on a recent description of the group of trivial-source endotrivial modules due to Grodal., Comment: 23 pages

In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$., Comment: 61 pages

We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group. Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves that all maximal subgroups with socle $\mathrm{PSL}_2(p^a)$ inside these finite almost simple groups are known, with three possible exceptions ($p^a=7,8,25$ for $E_7$). In the three remaining cases we provide considerable information about a potential maximal subgroup., Comment: v+154pp. Mem. Amer. Math. Soc., to appear

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

In 1998, Liebeck and Seitz introduced a constant $t(G)$, dependent on the root system of a reductive algebraic group $G$ and proved that if $x$ is a semisimple element of order greater than $t(G)$ in $G$ then there exists an infinite subgroup of $G$ stabilizing the same subspaces of $L(G)$ as $x$. The values for $t(G)$ are $12$, $68$, $124$ and $388$ for $G=G_2,F_4,E_6,E_7$ respectively. In this paper we obtain a similar result for these groups and the minimal module $V_{\mathrm{min}}$, obtaining significantly smaller numbers, namely $4$, $18$, $27$ and $75$ respectively (with some small conditions on the element $x$ that are not important for applications). Note that both $t(G)$ and these new bounds are sharp. As a corollary we eliminate several potential maximal subgroups $\mathrm{PSL}_2(q_0)$ of these groups that seem difficult to eliminate through other means, along with other groups. This paper forms part of the author's programme to vastly reduce the number of putative maximal subgroups of exceptional groups of Lie type., Comment: 10 pages

This article gives the structure of all projective indecomposable modules in blocks with non-cyclic defect group for $n\leq 11$, and almost all in the case of $n=12$, leaving four simple modules for $p=2$ and one simple module for $p=3$., Comment: 31 pages

In this paper we examine embeddings of alternating groups and symmetric groups into almost simple groups of exceptional type. In particular, we prove that unless the alternating or symmetric group has degree 6 or 7, there is no maximal subgroup of any almost simple group with socle an exceptional group of Lie type that is an alternating or symmetric group. Furthermore, in the remaining open cases we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aiming to substantially improve the state of knowledge about the maximal subgroups of exceptional groups of Lie type., Comment: 55 pages

In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Brou\'e's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Brou\'e's conjecture are consistent with both the mathematics and philosophy of blocks with cyclic defect groups., Comment: 15 pages

In this short note we prove that, if $p$ is an odd prime dividing the order of a sporadic simple group, then with the exception of four groups for $p=3$, all sporadic simple groups are generated by an involution and an element of order $p$.

We study Brou\'e's abelian defect group conjecture for groups of Lie type using the recent theory of perverse equivalences and Deligne--Lusztig varieties. Our approach is to analyze the perverse equivalence induced by certain Deligne--Lusztig varieties (the geometric form of Brou\'e's conjecture) directly; this uses the cohomology of these varieties, together with information from the cyclotomic Hecke algebra. We start with a conjecture on the cohomology of these Deligne--Lusztig varieties, prove various desirable properties about it, and then use this to prove the existence of the perverse equivalences predicted by the geometric form of Brou\'e's conjecture whenever the defect group is cyclic (except possibly for two blocks whose Brauer tree is unknown). This is a necessary first step to proving Brou\'e's conjecture in general, as perverse equivalences are built up inductively from various Levi subgroups. This article is the latest in a series by Raphael Rouquier and the author with the eventual aim of proving Brou\'e's conjecture for unipotent blocks of groups of Lie type., Comment: 56 pages, plus appendix

In this paper, we present a conjecture on the degree of unipotent characters in the cohomology of particular Deligne-Lusztig varieties for groups of Lie type, and derive consequences of it. These degrees are a necessary piece of data in the geometric version of Brou\'e's abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Brou\'e's conjecture should produce a more combinatorially defined derived equivalence, called a perverse equivalence. We prove that our conjectural degree is an integer (which is not obvious) and has the correct parity for a perfect isometry, and verify that it induces a perverse equivalence for all unipotent blocks of groups of Lie type with cyclic defect groups, whenever the shape of the Brauer tree is known (i.e., not E7 and E8). It has also been used to find perverse equivalences for some non-cyclic cases. This paper is a contribution to the conjectural description of the exact form of a derived equivalence proving Brou\'e's conjecture for groups of Lie type., Comment: This paper has now been superseded by arXiv:1207.0116: Pervese equivalences and Brou\'e's conjecture II: The cyclic case, and that article should be referenced for the results in this paper

In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups of Lie type in defining characteristic, the natural module is non-algebraic. For alternating and symmetric groups, we prove that the simple modules in $p$-blocks with defect groups of order $p^2$ are algebraic, for $p\leq 5$. Finally, we analyze nine sporadic groups, finding that all simple modules are algebraic for various primes and sporadic groups

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Brou\'e's conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups C3xC3. We also examine extensions to automorphism groups in a general setting., Comment: 50 pages

We introduce structure theorems for the study of the unit conjecture for supersoluble group rings and apply our results to the (Passman) fours group G. We show that over any field K, the group algebra KG has no non-trivial units of length at most 3, and find that the Promislow set can never be the support of a unit in KG. We conclude our work with an introduction to the theory of "consistent chains" toward a preliminary analysis of units of higher length in KG., Comment: 17 pages

In this article we study several classes of `small' 2-groups: we complete the classification, started in [Stancu, 2006], of all saturated fusion systems on metacyclic p-groups for all primes p. We consider Suzuki 2-groups, and classify all center-free saturated fusion systems on 2-groups of 2-rank 2. We end by classifying all possible F-centric, F-radical subgroups in saturated fusion systems on 2-groups of 2-rank 2., Comment: 22 pages, Submitted to Trans. Amer. Math. Soc

In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system.

This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups., Comment: 18 pages

In this article, we consider the control of fusion in fusion systems, proving three previously known, non-trivial results in a new, largely elementary way. We then reprove a result of Aschbacher, that the product of two strongly closed subgroups is strongly closed; to do this, we consolidate the theory of quotients of fusion systems into a consistent theory. We move on considering p-soluble fusion systems, and prove that they are constrained, allowing us to effectively characterize fusion systems of p-soluble groups. This leads us to recast Thompson Factorization for Qd(p)-free fusion systems, and consider Thompson Factorization for more general fusion systems., Comment: 24 pages

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if L is a component of the (stable) Auslander-Reiten quiver for a dihedral 2-group consisting of non-periodic modules, then there is at most one algebraic module on L., Comment: 11 pages

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander--Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity., Comment: 10 pages

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having this property is equivalent to the tensor structure being 'nice' for that module. In this paper we prove that if G is a group with abelian Sylow 2-subgroups, and p=2, then all simple modules for G are algebraic. We include the conjecture that this result holds for all abelian 2-blocks., Comment: 9 pages

In this article we prove the following result: that for any two natural numbers k and j, and for all sufficiently large symmetric groups Sym(n), there are k disjoint sets of j irreducible characters of Sym(n), such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moret\'o. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature., Comment: 24 pages, to appear in Proc. London Math. Soc