A combinatorial condition on a certain variety of groups

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R Δ(F) is solvable.

Combinatorial Conditions Forcing Commutativity of an Infinite Group

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.

Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(S-quasinormal)-group if every local subgroup of G is either an S-quasinormal subgroup or conjugate to a subgroup of S. The main purpose of this construction is to demonstrate a new way of analyzing the structure of a finite group by the properties and the number of conjugacy classes of its local subgroups.

A proper subgroup M of a group G is called a CC-subgroup of G if the centralizer C G (m) of every m ∈ M # = M ∖ {1} is contained in M. In this paper we classify all finite groups containing a CC-subgroup, extending work of many authors.

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in Rn singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

Strong involutions in finite special linear groups of odd characteristic

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnicket al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019,arXiv:1904.08070] by different methods.

A class of finite groups with abelian 2-Sylow subgroups

In a previous paper the author proposed a representation r for the algebra of representations of a finite group G. Here a corresponding representation c for the algebra of classes of G is established, and the relations between the two representations are investigated. In particular, the idempotents of r and c are studied in some detail.

Let \(\mathfrak{H}\) be a class of finite groups. An \(\Omega \)-fiber formation \(\mathfrak{F}\) of finite groups with direction \(\varphi \) is said to be a minimal \(\Omega \)-fiber non-\(\mathfrak{H}\)-formation with direction \(\varphi \), or briefly an \(\mathfrak{H}_\Omega \)-critical formation, if \(\mathfrak{F} \nsubseteq \mathfrak{H}\), but any proper \(\Omega \)-fiber subformation with direction \(\varphi \) in \(\mathfrak{F}\) belongs to the class \(\mathfrak{H}\). In the paper, a complete description of the structure of minimal \(\Omega \)-fiber non-\(\mathfrak{H}\)-formations of finite groups of two different directions is given.

We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.