Narrow Results

A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set X, such that every element of it can be written as a palindrome in the letters of X. Moreover, every simple group has palindromic width pw(G, X) = 1, where X only differs by at most one additional generator from any given generating set. On the contrary, we prove that all non-abelian finite simple groups G also have a generating set S with pw(G, S) > 1. As a by-product of our work we also obtain that every just-infinite group has finite palindromic width with respect to a finite generating set. This provides first examples of groups with finite palindromic width but infinite commutator width.

We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either $\{x,y\}$ is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m₂(G) is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp₄ and G₂. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context which is substantially broader than the K* setting.

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

Let G be a finite group and let cd(G) be the set of all irreducible complex character degrees of G. It was conjectured by Huppert in Illinois J. Math.44 (2000) that, for every non-abelian finite simple group H, if cd(G) = cd(H) then G ≅ H × A for some abelian group A. In this paper, we confirm the conjecture for the family of projective special linear groups PSL₄(q) with q ≥ 13.

In this paper, we prove that every element in the alternating group $A_n$, with $n\ge 5$, can be written as a product of at most two Engel words of arbitrary length.

We investigate a possible connection between the $FSZ$ properties of a group and its Sylow subgroups. We show that the simple groups $G_2(5)$ and $S_6(5)$, as well as all sporadic simple groups with order divisible by $5^6$ are not $FSZ$, and that neither are their Sylow 5-subgroups. The groups $G_2(5)$ and $HN$ were previously established as non-$FSZ$ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be $FSZ$. We conclude by considering all perfect groups available through GAP with order at most $10^6$, and show they are non-$FSZ$ if and only if their Sylow 5-subgroups are non-$FSZ$.

In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group $G$ with this property is an extension of an almost simple group $S$ by a solvable group and $|\rho (G)|\le |\rho (S)|+2$, where $\rho (G)$ is the set of all primes dividing some character degree of $G$.

In this paper, the non-abelian simple groups having at most 20 distinct irreducible character degrees are classified.

A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|,|K|)=1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|)=1. In this paper, we classify all finite non-abelian simple groups which contain a non-trivial semipermutable (s-semipermutable) subgroup. As a corollary of our main result, we give a complete answer to an unsolved problem in group theory proposed by V.S. Monakhov in 1990.

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut(X) acts transitively on the vertices and the vertex-stabilizer A_v of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

The probability that a finite group G is generated by s elements is given by a truncated Dirichlet series in s, denoted by P_G(s). We give an explicit criterion that allows one to recognize whether the factor group G/Frat(G) is simple by only looking at the coefficients of P_G(s). In order to get such a criterion, we prove that the series derived from P_G(s) by removing the even-indexed terms has only a simple zero at s=1.

In this short note we prove that the finite non-abelian simple groups PSL(2, q), where q = 5, 7, are determined by their posets of classes of isomorphic subgroups. Several interesting open problems are also formulated.

We aim to study maximal pairwise commuting sets of 3-transpositions (transvections) of the simple unitary group $U_n(2)$ over $\mathrm{GF}(4)$, and to construct designs from these sets. Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions. Let $G=U_n(2)$. It is well known that $G$ is a 3-transposition group with the set $D$, the conjugacy class consisting of its transvections, as the set of 3-transpositions. Let $L$ be a set of basic transpositions in $D$. We give general descriptions of $L$ and $1$- $(v,k,\lambda )$ designs $\mathcal{D}=(\mathcal{P},\mathcal{B})$, with $\mathcal{P}=D$ and $\mathcal{B}=\{L^g\mid g\in G\}$. The parameters $k=|L|$, $\lambda$ and further properties of $\mathcal{D}$ are determined. We also, as examples, apply the method to the unitary simple groups $U_4(2)$, $U_5(2)$, $U_6(2)$, $U_7(2)$, $U_8(2)$ and $U_9(2)$.

The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper surveys methods and results relating the Alperin weight conjecture, especially the recent developments in the inductive investigation.

A finite permutation group $G\le \mathrm{\text{Sym}}(\mathrm{\Omega })$ is called $2$-closed if $G$ is the largest subgroup of $\mathrm{\text{Sym}}(\mathrm{\Omega })$ which leaves invariant each of the $G$-orbits for the induced action on $\mathrm{\Omega }\times \mathrm{\Omega }$. Introduced by Wielandt in 1969, the concept of $2$-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total $2$-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group $G$ is said to be totally $2$-closed if $G$ is $2$-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally $2$-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly $6$ totally $2$-closed finite nonabelian simple groups: the Janko groups ${\mathrm{\text{J}}}_1$, ${\mathrm{\text{J}}}_3$ and ${\mathrm{\text{J}}}_4$, together with $\mathrm{\text{Ly}}$, $\mathrm{\text{Th}}$ and the Monster $𝕄$. Moreover, if a finite totally $2$-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely $47$ examples.

In the course of obtaining this classification, we develop a general framework for studying $2$-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Let $G$ be a finite group, whose order has $n$ prime divisors. In this paper, we prove that if $G$ has a $p$-complement for $n-1$ prime divisors $p$ of $\left|G\right|$ and $G$ has no section isomorphic to $L_2(7)$. Then $G$ is solvable, which generalizes a theorem of Hall.

We present some comments about the least common multiple of the sizes of conjugacy classes. It is noted how the problems related to this least common multiple connect to questions concerning the existence of regular orbits of linear groups.