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We study embeddings of groups of Lie type With a few possible exceptions, we prove that there are no Lie primitive subgroups We prove a slightly stronger result, including stability under automorphisms of The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.

This article produces a complete list of all maximal subgroups of the finite simple groups of type F4, E6 and twisted E6 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 F42(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 E6. We provide a collection of supplementary 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.

In this article we study several classes of 'small' 2-groups: we complete the classification, started by Stancu, 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.

In this article we study several classes of ‘small’ 22-groups: we complete the classification, started by Stancu, of all saturated fusion systems on metacyclic pp-groups for all primes pp. We consider Suzuki 22-groups, and classify all center-free saturated fusion systems on 22-groups of 22-rank 22. We end by classifying all possible F\\mathcal {F}-centric, F\\mathcal {F}-radical subgroups in saturated fusion systems on 22-groups of 22-rank 22.

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.

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}\\).)

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.

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\\).

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.