Frank Wagner
Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal quotient. In the supersimple case, the Berline-Lascar decomposition works. One-based simple groups are finite-by-abelian-by-finite.
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In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.
Externally definable fsg groups in NIP theories
本文证明NIP结构中外部可定义的fsg群与可解释群可定义同构，利用诚实定义与群块重构方法，并推广了外部可定义集合与群的相关结果，包括证明Eleftheriou猜想。
In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.
A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent.
In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are $\mathbb{Z}_3$ and $S_3$. We also show that the only nonsolvable group with two supercharacter theories is ${\rm Sp} (6,2)$.
We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G^{00}, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M^{ext}, of a model M. In the light of these results we continue the study of the "definable topological dynamics" of groups in NIP theories. In particular we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in some new cases, including definably amenable groups in o-minimal structures.
We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The first-one concerns the non-superstability of such groups in most cases, and the second-one concerns the connectedness of most free products of groups.
In 2008, Diaconis annd Isaacs introduced the notion of a supercharacter theory of a finite group in which supercharacters replace with irreducible characters and superclasses by conjugacy classes. In this paper, we introduce an algorithm for constructing supercharacter theories of a finite group by which all supercharacter theories of groups containing up to 14 conjugacy classes are calculated.
Much can be learned about a finite group from its character table, but sometimes that table can be difficult to compute. Supercharacter theories are generalizations of character theory defined by P. Diaconis and I.M. Isaacs, in which certain (possibly reducible) characters called supercharacters take the place of the irreducible characters, and a certain coarser partition of the group takes the place of the conjugacy classes. We present five new ways to construct new supercharacter theories out of supercharacter theories already known to exist, including a direct product, a lattice-theoretic j…
A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular elements of $G$ that satisfy certain properties. We classify the groups and primes that have exactly one super-Brauer character theory. We discuss the groups with exactly two super-Brauer character theories.
A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions.The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.
The Jordan-Hölder Theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan-Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan-Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.
We describe the links between group theory and psychology, in particular through the works of Piaget. We show that groups appear universally in his description of children's intelligence, and that the notion of groupoid, which was little considered in psychology, may be fundamental. We study in particular the applicability of group theory concepts to the development of educative games.
Cohomology Theories of Partial Groups
本文提出部分群的上同调理论，通过两种方法证明其一致性，并应用该理论计算部分群扩展，对比经典群扩展结果。
We describe the supercharacter theories of the semidirect product of H and K, $H\rtimes K$ in terms of the supercharacter theories of the direct product of H and K in the case when both H and K are Abelian groups. To do this we introduce the concept of a homomorphism of supercharacter theories. This provides a classification of the supercharacter theories of the dihedral groups of order 2m when m is odd using the known classification of the supercharacter theories of cyclic groups.
We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial two-dimensional surfaces to the category of Hilbert spaces and bounded linear operators.
Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain fini…
We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the category hermitian representations, as an $\infty$-category, is equivalent to a category of usual representations.
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