Group Theory Papers of the Week

In this series of two papers, including Li et al. [Proc.

We prove the Díaz$\unicode[Arial]{x2013}$Park conjecture for sharpness of saturated fusion systems.

We prove that the braided Thompson groups $BV$ and $BF$ are Hopfian, answering the two parts of Zaremsky's question.

We study the random walk on a Cayley graph of a finite group whose generators are chosen independently and uniformly at random.

We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field.

We prove every Cayley graph of odd order satisfies the conjecture.

We prove that $\varphi$ has at most $|Q:Q'|$ lifts to ordinary irreducible characters, establishing Cossey's conjecture for odd primes.

We propose a new refinement of the (blockwise) Alperin weight conjecture taking dimensions of simple modules (or degrees of irreducible Brauer characters) into consideration.

We prove that if every maximal subgroup of $G$ of order divisible by $p$ containing the normalizer of some Sylow subgroup is nilpotent then $G$ is a solvable group having a Sylow tower and $G$ is $q$-nilpotent or $q$-closed for each prime divisor $q$ of $|G|$.

We prove that the intersection of all non-nilpotent maximal subgroups of $G$ of order divisible by $p$ containing the normalizer of some Sylow subgroup is $p$-closed, which provides a further generalization of Shidov's theorem and Shlyk's theorem.

We characterize finite groups whose involutions form a single fusion class.

We construct a ZP-resolution Sh_P of St_P, the P-sharbly resolution.

In this paper, let $AM(G)$ denote the number of conjugacy classes of all almost maximal subgroups of $G$.

Abstract A strong area of GAP has always been the character and representation theory of (mostly finite) groups. In this survey we will highlight the functionality of GAP in this area and mention some example applications.

Huber (2010) reduced the Steiner case ($\lambda=1$) to a single infinite family of possible exceptions, where the automorphism group is $G=\PGaL(2,p^e)$ acting on the points of the projective line, with $p\in\{2,3\}$ and $e$ an odd prime power. In this paper, we resolve this final case.

We construct explicit two-sided lossless expanders whose binary adjacency matrices have full rank, and use them to obtain explicit asymptotically good quantum LDPC codes whose specified X- and Z-check matrices have full row rank and satisfy small-set soundness, or confinement, on a linear range.

We introduce the notion of Boomerang subalgebras of group von Neumann algebras and study their general properties.

Given a non-solvable-by-finite subgroup G of GL(d, F), where F is the field of complex numbers or a non-archimedean local field, we present methods to construct explicitly a non-abelian 2-generator free subgroup of G in specific cases.