Related papers: A note on Oliver's p-group conjecture

Sendov's conjecture, which was first introduced in the last 50s, asserts that if all the zeros of a polynomial $p$ lie in the closed unit disk then for each zero there must be a critical point of $p$ within unit distance. This paper…

Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p)…

Let A be a simple, unital, exact, and finite C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup obtained from the Elliott invariant…

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is…

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that…

Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of…

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

We prove the Andre-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n be a positive integer, and let…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all $i\ge 0$: (i) $\exp \Omega_-i(G)\le p^i$ for odd p, and $\exp \Omega_-i(G)\le 2^{i+1}$ for p = 2; (ii) the index $|G:G^{p^i}|$…

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely…

A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to…

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…

We prove Gieseker conjecture for an homogeneous space $X$, saying that if $X$ has no non-trivial tame coverings then it has no non-trivial regular singular $\mathscr{O}_X$-coherent $\mathscr{D}_{X/k}$-modules. In order to do so we prove a…

In this short note we answer to a question of group theory from arXiv:0910.5080. In that paper the author describes the set of realizable Steinitz classes for so-called $A'$-groups of odd order, obtained iterating some direct and semidirect…

Given a Lie algebra $\mathfrak{g}$ of an algebraic group over a ring $S,$ we show that the first Kac-Weisfeiler conjecture holds for reductions of $\mathfrak{g} \mod p$ for large enough primes $p,$ reproving a recent result of Martin,…

For any odd prime $p$ and any integer $n>0$ with $p^2|n$, we show that the mod $p$ cohomology ring of the classifying space of the projective unitary group $PU(n)$ is not completely detected by elementary abelian $p$-subgroups, providing…

Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Sch\"onheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be…

In this note we give a particularly short and simple proof of the following theorem of Karrass and Solitar. Let $H$ be a finitely generated subgroup of a free group $F$ with infinite index $[F:H]$. Then there is a nontrivial normal subgroup…

In 2017, Jones studied the unitary representations of Thompson's group $F$ and defined a method to construct knots and links from $F$. One of his results is that any knot or link can be obtained from an element of this group, which is…