Diameters of finite simple groups: sharp bounds and applications

@article{Liebeck2001DiametersOF,
  title={Diameters of finite simple groups: sharp bounds and applications},
  author={Martin W. Liebeck and Aner Shalev},
  journal={Annals of Mathematics},
  year={2001},
  volume={154},
  pages={383-406}
}
Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph r(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of arbitrary order). We also show that for any word w = w(xl,..., xd), there is a constant c = c(w) such that for any simple group G on which w does… 
Width questions for finite simple groups
This thesis is concerned with the study of non-abelian finite simple groups and their generation. In particular, we are interested in a class of problems called width questions which measure the rate
On the diameters of McKay graphs for finite simple groups
Let G be a finite group, and α a nontrivial character of G . The McKay graph $${\cal M}\left({G,\alpha } \right)$$ ℳ ( G , α ) has the irreducible characters of G as vertices, with an edge from χ 1
An improved diameter bound for finite simple groups of Lie type
For a finite group G , let diam (G) denote the maximum diameter of a connected Cayley graph of G . A well‐known conjecture of Babai states that diam (G) is bounded by (log2|G|)O(1) in case G is a
A diameter bound for finite simple groups of large rank
TLDR
If $G$ has rank $n$, and its base field has order $q$, then the diameter of $\Gamma (G,S)$ would be $q^{O(n(\log n + \log q)^3)}$.
On finitely generated profinite groups, I: strong completeness and uniform bounds
We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is
On finitely generated profinite groups, II: products in quasisimple groups
We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted
Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
  • H. Helfgott
  • Mathematics
    Groups St Andrews 2017 in Birmingham
  • 2019
By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on
On the product decomposition conjecture for finite simple groups
We prove that if G is a finite simple group of Lie type and S is a subset of G of size at least two, then G is a product of at most c log|G|/log |S| conjugates of S, where c depends only on the Lie
...
...

References

SHOWING 1-10 OF 34 REFERENCES
Upper Bounds for the Number of Conjugacy Classes of a Finite Group
Abstract For a finite group G , let k ( G ) denote the number of conjugacy classes of G . We prove that a simple group of Lie type of untwisted rank l over the field of q elements has at most (6 q )
On the Diameter of a Cayley Graph of a Simple Group of Lie Type Based on a Conjugacy Class
TLDR
The conjugacy diameter cd(G) of G is defined to be the maximum diameter of all such Cayley graphs as C ranges over all non-identity conjugate classes.
Products of powers in finite simple groups
LetG be a group. For a natural numberd≥1 letGd denote the subgroup ofG generated by all powersad,a∈G.A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated
A note on powers in simple groups
In [ 7 ], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S . The motivation for proving this
Probabilistic generation of finite simple groups, II
Varieties and simple groups
  • G. Jones
  • Mathematics
    Journal of the Australian Mathematical Society
  • 1974
In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “ Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple
On the Minimal Degrees of Projective Representations of the Finite Chevalley Groups
For G = G(q), a Chevalley group defined over the field iFQ of characteristic p, let Z(G,p) be th e smallest integer t > 1 such that G has a projective irreducible representation of degree t over a
Upper bound on the characters of the symmetric groups
Abstract Let C be a conjugacy class in the symmetric group Sn, and λ be a partition of n. Let fλ be the degree of the irreducible representation Sλ, χλ(C)– the character of Sλ at C, and rλ(C)– the
Local Expansion of Symmetrical Graphs
TLDR
The expansion rate of a subset S of the vertex set is the quotient e(S), where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S .
Covering Theorems for FINASIGS VIII—almost all conjugacy classes in A n have exponent ≤4
Abstract The product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is
...
...