Finite Nonabelian Simple Group Research Articles

Chiral polyhedra and finite simple groups

We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$ and $A_7$.

Pentavalent symmetric graphs admitting vertex-transitive non-abelian simple groups

A graph Γ is said to be symmetric if its automorphism group Aut(Γ) is transitive on the arc set of Γ. Let G be a finite non-abelian simple group and let Γ be a connected pentavalent symmetric graph with G≤Aut(Γ). In this paper, we show that if G is transitive on the vertex set of Γ, then either G⊴Aut(Γ) or Aut(Γ) contains a non-abelian simple normal subgroup T such that G≤T and (G,T) is one of 58 possible pairs of non-abelian simple groups. In particular, if G is transitive on the arc set of Γ, then (G,T) is one of 17 possible pairs, and if G is regular on the vertex set of Γ, then (G,T) is one of 13 possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, et al. (2011).

Free subgroups of inverse limits of iterated wreath products of non-abelian finite simple groups in primitive actions

Abstract Let 𝒲 = { G i ∣ 1 ≤ i ∈ ℕ } {\mathcal{W}=\{G_{i}\mid 1\leq i\in\mathbb{N}\}} be a set of non-abelian finite simple groups. Set W 1 = G 1 {W_{1}=G_{1}} and choose a faithful transitive primitive W 1 W_{1} -set Δ 1 \varDelta_{1} . Assume that we have already constructed W n - 1 W_{n-1} and chosen a transitive faithful primitive W n - 1 W_{n-1} -set Δ n - 1 \varDelta_{n-1} . The group W n W_{n} is then defined as W n = G n ⁢ wr Δ n - 1 ⁡ W n - 1 {W_{n}=G_{n}\operatorname{wr}_{\varDelta_{n-1}}W_{n-1}} . If W is the inverse limit W = lim ← ⁡ ( W n , ρ n ) {W=}{\varprojlim(W_{n},\rho_{n})} with respect to the natural projections ρ n : W n → W n - 1 {\rho_{n}\colon W_{n}\to W_{n-1}} , we prove that, for each k ≥ 2 k\geq 2 , the set of k-tuples of W that freely generate a free subgroup of rank k is comeagre in W k W^{k} and its complement has Haar measure zero.

Generation of finite simple groups by an involution and an element of prime order

We prove that every non-abelian finite simple group is generated by an involution and an element of prime order.

A diameter bound for finite simple groups of large rank

Given a non-abelian finite simple group $G$ of Lie type, and an arbitrary generating set $S$, it is conjectured by Laszlo Babai that its Cayley graph $\Gamma (G,S)$ will have a diameter of $(\log |G|)^{O(1)}$. However, little progress has been made when the rank of $G$ is large. In this article, we shall show that if $G$ has rank $n$, and its base field has order $q$, then the diameter of $\Gamma (G,S)$ would be $q^{O(n(\log n + \log q)^3)}$.

Lie theory of finite simple groups and the Roth property

AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

Finite almost simple groups with prime graphs all of whose connected components are cliques

We find finite almost simple groups with prime graphs all of whose connected components are cliques, i.e., complete graphs. The proof is based on the following fact, which was obtained by the authors and is of independent interest: the prime graph of a finite simple nonabelian group contains two nonadjacent odd vertices that do not divide the order of the outer automorphism group of this group.

Mersenne primes and solvable Sylow numbers

Let [Formula: see text] be a prime. The Sylow [Formula: see text]-number of a finite group [Formula: see text], which is the number of Sylow [Formula: see text]-subgroups of [Formula: see text], is called solvable if its [Formula: see text]-part is congruent to [Formula: see text] modulo [Formula: see text] for any prime [Formula: see text]. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow [Formula: see text]-number of a finite group is the product of two kinds of factors: of prime powers [Formula: see text] with [Formula: see text] (mod [Formula: see text]) and of the number of Sylow [Formula: see text]-subgroups in certain finite simple groups (involved in [Formula: see text]). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to [Formula: see text] for [Formula: see text] a Mersenne prime.

The inductive blockwise Alperin weight condition for G2(q) and [formula omitted

The inductive blockwise Alperin weight condition is a system of conditions whose verification for all non-abelian finite simple groups would imply the blockwise Alperin weight conjecture. We establish this condition for the groups G2(q), q⩾5, and D43(q) for all primes dividing their order.

Double quadrics with large automorphism groups

We classify nodal Fano threefolds that are double covers of smooth quadrics branched over intersections with quartics and are acted on by finite simple non-abelian groups. We also study their rationality.

Finite simple groups in which all maximal subgroups are π-closed. I

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph ΔG of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.

A 2-ARC TRANSITIVE PENTAVALENT CAYLEY GRAPH OF

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

On pentavalent 1-transitive Cayley graphs

A Cayley graph Γ=Cay(G,S) is said to be core-free if G is core-free in some X≤AutΓ. In this paper, a characterization of connected core-free pentavalent 1-transitive Cayley graphs is given. Moreover, the argument in this paper also gives another proof for a result in Zhou and Feng (2010) which says that all connected pentavalent 1-transitive Cayley graphs of finite non-abelian simple groups are normal.

The inductive Alperin–McKay condition for 2-blocks with cyclic defect groups

We verify the inductive Alperin–McKay condition introduced by the second author, for 2-blocks of the covering groups of finite simple non-abelian groups with cyclic defect groups.

CONSTRUCTION OF UNRAMIFIED EXTENSIONS WITH A PRESCRIBED GALOIS GROUP

In this article, we shall prove that for any finite solvable group $G$, there exist infinitely many abelian extensions $K/\mathbb{Q}$ and Galois extensions $M/\mathbb{Q}$ such that the Galois group $\Gal(M/K)$ is isomorphic to $G$ and $M/K$ is unramified. The difference between our result and [3, 4, 6, 7, 13] is that we have a base field $K$ which is not only Galois over $\mathbb{Q}$, but also has very small degree compared to their results. We will also get another proof of Nomura's work [9], which gives us a base field of smaller degree than Nomura's. Finally for a given finite nonabelian simple group $G$, we will show there exists an unramified extension $M/K'$ such that the Galois group is isomorphic to $G$ and $K'$ has relatively small degree.

Finite simple groups of low rank: Hurwitz generation and (2,3)-generation

‎Let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $F$ of characteristic $pgeq 0$‎. ‎We survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions.

Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.

The Ore conjecture

The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character theoretic methods with other ingredients, and use them to establish the conjecture. Liebeck acknowledges the support of a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Applications. O’Brien acknowledges the support of an LMS Visitor Grant, the Marsden Fund of New Zealand (grant UOA 0721), and the Mathematical Sciences Research Institute (Berkeley). Shalev acknowledges the support of an EPSRC Visiting Fellowship, an Israel Science Foundation Grant, and a Bi-National Science Foundation grant United States-Israel 2004-052. Tiep acknowledges the support of the NSF (grant DMS-0600967), and the Mathematical Sciences Research Institute (Berkeley).