Related papers: The Amit-Ashurst conjecture for finite metacyclic …

The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Sp\"ath showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all…

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that if $G$ is an odd order finite non-abelian monolithic $p$-group such…

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should include all…

I investigate modal group theory for arbitrary homomorphisms. Possibility is interpreted by the existence of a group homomorphism out of the given group, so the semantics is governed by the possibility of collapse: elements may be…

We prove that the cut space of any transitive graph $G$ is a finitely generated ${\rm Aut}(G)$-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose…

A $p$-group $G$ is called *ab-maximal* if $|H : H'| < |G:G'|$ for every proper subgroup $H$ of $G$. Similarly, $G$ is called *$d$-maximal* if $d(H) < d(G)$ for every proper subgroup $H$ of $G$, where $d(H)$ is the minimal number of…

Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps $\mathbb Z^2 \rightarrow \mathcal A$ where $\mathcal A$ is a finite set of symbols. Such configurations are often…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

Let $N$ be a nilpotent normal subgroup of the finite group $G$. Assume that $u$ is a unit of finite order in the integral group ring $\mathbb{Z} G$ of $G$ which maps to the identity under the linear extension of the natural homomorphism $G…

For a finite group $G$, we denote by ${\sf d}(G)$ and by ${\sf E}(G)$, respectively, the small Davenport constant and the Gao constant of $G$. Let $C_n$ be the cyclic group of order $n$ and let $G_{m,n,s} = C_n \rtimes_s C_m$ be a…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…

Let $\chi$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $\chi(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of…

For any positive integers m and n, the word map (x,y) -> x^m y^n is almost measure preserving on large finite simple groups G.

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley,…

We consider the probability theory, and in particular the moment problem and universality theorems, for random groups of the sort of that arise or are conjectured to arise in number theory, and in related situations in topology and…

For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and…

According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. There is a recent interest to generalize…