Alternating and symmetric groups with Eulerian generating graph

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\Gamma(G)$ when $G$ is an alternating group or a symmetric group. In particular, we determine the vertices of $\Gamma(G)$ having even degree and show that $\Gamma(G)$ is Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4

Deformation theory and finite simple quotients of triangle groups I

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $\mu=1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$

Triangle groups and finite simple groups

This thesis contains a study of the spaces of homomorphisms from hyperbolic triangle groups to finite groups of Lie type which leads to a number of deterministic, asymptotic, and probabilistic results on the (p1, p2, p3)-generation problem for finite groups of Lie type. Let G0 = L(pn) be a finite simple group of Lie type over the finite field Fpn and let T = Tp1,p2,p3 be the hyperbolic triangle group (x,y : xp1 = yp2 = (xy)p3 = 1) where p1, p2, p3 are prime numbers satisfying the hyperbolic condition 1/p1 + 1/p2 + 1/p3 < 1. In general, the size of Hom(T,G0) is a polynomial in q, where q = pn, whose degree gives the dimension of Hom(T,G), where G is the corresponding algebraic group, seen as a variety. Computing the precise size of Hom(T,G0) or giving an asymptotic estimate leads to a number of applications. One can for example investigate whether or not there is an epimorphism in Hom(T,G0). This is equivalent to determining whether or not G0 is a (p1, p2, p3)-group. Asymptotically, one might be interested in determining the probability that a random homomorphism in Hom(T,G0) is an epimorphism as |G0| → ∞. Given a prime number p, one can also ask wether there are finitely, or infinitely many positive integers n such that L(pn) is a (p1, p2, p3)-group. We solve these problems for the following families of finite simple groups of Lie type of small rank: the classical groups PSL2(q), PSL3(q), PSU3(q) and the exceptional groups 2B2(q), 2G2(q), G2(q), 3D4(q). The methods involve the character theory and the subgroup structure of these groups. Following the concept of linear rigidity of a triple of elements in GLn(Fp), used in inverse Galois theory, we introduce the concept for a hyperbolic triple of primes to be rigid in a simple algebraic group G. The triple (p1, p2, p3) is rigid in G if the sum of the dimensions of the subvarieties of elements of order p1, p2, p3 in G is equal to 2 dim G. This is the minimum required for G(pn) to have a generating triple of elements of these orders. We formulate a conjecture that if (p1, p2, p3) is a rigid triple then given a prime p there are only finitely many positive integers n such that L(pn) is a (p1, p2, p3)-group. We prove this conjecture for the classical groups PSL2(q), PSL3(q), and PSU3(q) and show that it is consistent with the substantial results in the literature about Hurwitz groups (i.e. when (p1, p2, p3) = (2, 3, 7)). We also classify the rigid hyperbolic triples of primes in algebraic groups, and in doing so we obtain some new families of non-Hurwitz groups

On finite groups with the Magnus Property

We investigate finite groups with the Magnus Property, where a group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse conjugate. In particular we observe that a finite MP group is solvable, determine the finite primitive MP groups and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown-based powers of a finite monolithic primitive group

On irreducible subgroups of simple algebraic groups

Let G be a simple algebraic group over an algebraically closed field K of characteristic p &gt; 0, let H be a proper closed subgroup of G and let V be a nontrivial irreducible KG-module, which is p-restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples (G, H, V ) of this form when G is a classical group and H is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples (G, H, V ) to the case where G is an orthogonal group, V is a spin module and H normalizes an orthogonal decomposition of the natural KG-module

The pro-kk-solvable topology on a free group

We prove that, given a finitely generated subgroup $H$ of a free group $F$, the following questions are decidable: is $H$ closed (dense) in $F$ for the pro-(met)abelian topology? is the closure of $H$ in $F$ for the pro-(met)abelian topology finitely generated? We show also that if the latter question has a positive answer, then we can effectively construct a basis for the closure, and the closure has decidable membership problem in any case. Moreover, it is decidable whether $H$ is closed for the pro-${\bf V}$ topology when ${\bf V}$ is an equational pseudovariety of finite groups, such as the pseudovariety ${\bf S}_k$ of all finite solvable groups with derived length $\leq k$. We also connect the pro-abelian topology with the topologies defined by abelian groups of bounded exponent