Related papers: Groups with essential dimension one

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

This is the third one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups in odd dimension.

We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$…

In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This…

Fix $\varepsilon > 0$. We say that a finite group $G$ is $\varepsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $|G|^\varepsilon$. In this paper, we give a structure theorem for large…

We describe the fundamental groups of ordered and unordered k point sets in complex projective space of dimension n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of…

We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.

The essential dimension $\operatorname{ed}_k({\rm S}_n)$ of the symmetric group ${\rm S}_n$ is the minimal integer $d$ such that the general polynomial $x^n + a_1 x^{n-1} + \ldots + a_n$ can be reduced to a $d$-parameter form by a…

We determine the irreducible representations of alternating and symmetric groups and their universal central extensions that contain a non-scalar element with all but one eigenvalues of multiplicity 1. The ground field is algebraically…

Let K be any field and G be a finite group. Noether's problem asks whether the fixed field is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p^n containing a cyclic subgroup of index p…

Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.

In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups…

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the…

We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups)…

We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition…

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.