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the first nine parts of a series devoted to simplify the proof of the Classification of finite simple groups; in the spirit of preserving the (if not the original hundreds of papers of the 20th century) the actual proof of the theorem

We show that the only finite simple groups of essential dimension 3 (over C) are A_6 and possibly PSL(2,11). This is an easy consequence of the classification by Prokhorov of rationally connected threefolds with an action of a simple group.

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group A_5 in dimensions 2, 3 and 5, the linear fractional group PSL_2(7) in dimension 5, and possibly the unitary group PSU_3(3) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.

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 $q^{O(n(\log n + \log q)^3)}$.

The classification of finite simple strongly real groups is complete. It is easy to see that strong reality for every nonabelian finite simple group is equivalent to the fact that each element can be written as a product of two involutions. We thus obtain a solution to Problem 14.82 from the Kourovka notebook from the classification of finite simple strongly real groups.

We prove that if G is SL_2(F) or PSL_2(F), where F is a finite field, and A is a set of generators of G, then either |AAA| > |A|^(1+epsilon), where epsilon is an absolute positive real number, or AAA=G. As a corollary we get that the diameter of any Cayley graph of G is Poly-Logarithmic in |G|.

For most (and possibly all) non-associative finite simple Moufang loops, three generators of order 3 can be chosen so that each two of them generate a group isomorphic to $(3, 3 | 3, p)$. The subgroup structure of $(3, 3 | 3, p)$ depends on the solvability of a certain quadratic congruence, and it is described here in terms of generators.

We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.

We prove that if $G$ is a finite simple group of Lie type and $S$ 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 rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.

We prove that if $G$ is a finite simple group of Lie type and $S$ 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 rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.

We prove that Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of "Kourovka notebook".

This paper has been withdrawn since it is superated by the latest version of arXiv:0812.1139 (this is the version which will appear in Rend. Circ. Mat. Palermo; it is a strengthening and elaboration of a paper published in Math. Proc. Camb. Phil. Soc. 144, 397-401 (2008))

This paper has been withdrawn since it is superated by the latest version of arXiv:0812.1139 (this is the version which will appear in Rend. Circ. Mat. Palermo; it is a strengthening and elaboration of a paper published in Math. Proc. Camb. Phil. Soc. 144, 397-401 (2008))

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.

We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSL_n(q) and PSp_{2n}(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSU_n(2^m).

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5.

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.

Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter {\omega} on N and a sequence {(G_i)_{i \in N}} of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the {\omega}-limit of the characteristics of the groups Gi. Moreover G is a complete path-connected group with respect to the natural metric on G.

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $\mu=1/a+1/b+1/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 $T$.

Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.

Let ${\rm GK}(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that $|H|=|G|$ and $D(H)=D(G)$. A 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this paper is threefold. First, it provides the reader with a few useful and efficient tools on OD-characterizability of finite groups. Second, it lists a number of such simple groups that have been already investigated. Third, it shows that the simple groups $L_6(3)$ and $U_4(5)$ are OD-characterizable, too.

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ if and only if (i) $A$ contains the identity and (ii) $A$ is invariant under $\Aut (G)$.

Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $C$ such that $\dim H^2(G,M)\leq C\dim M$ for every finite simple group $G$, every prime $p$ and every irreducible $F_p [G]$-module $M$.

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.

Finite simple groups of Lie type as expanders

The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial.

We apply spectral analysis of quotients of the Bruhat-Tits buildings of type $\tilde{A}_{d-1}$ to construct isospectral non-isomorphic Cayley graphs of the finite simple groups $\operatorname{PSL}_d({\mathbb F}_q)$ for every $d \geq 5$ ($d \neq 6$) and prime power $q > 2$.

Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except $J_2$, $A_6$, $A_{10}$ and $^3D_4(2)$, are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: there exists a positive integer $d_0$ such that every finite simple classical group of dimension larger than $d_0$ is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a~finite group isospectral to a finite simple symplectic or orthogonal group $L$ of dimension at least 10, is either isomorphic to $L$ or not a group of Lie type in the same characteristic as $L$, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with $d_0=60$.

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 $A_7$.

Here we study the automorphism groups of $1$-designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, 2011). We obtain some general results for both and improve one of these methods. In an application to the sporadic Mathieu groups $M_{n}$, we are able to retrieve the Steiner systems $S(t,t+3,n)$ where $(n,t)\in\{(22,3),(23,4),(24,5)\}$.

We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).

We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).

In 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 C stable under inversion. We study the associated Killing form. For the universal calculus associated to C=G \ {e} we show that the magnitude of the Killing form \mu=\sum_{a,b\in C}K^{-1}_{a,b} 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 \mu\ is not equal to 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 almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known that most finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the generating conjugacy class in the case of the dihedral groups D_{2n} with n odd has invertible Killing form, and the same for the 2-cycles conjugacy class in any S_n. We also compute some eigenvalues of the Killing form in the case of the n-cycles class in S_n. Finally, 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.

Let $H_k(W,q)$ be the Iwahori--Hecke algebra associated with a finite Weyl group $W$, where $k$ is a field and $0 \neq q \in k$. Assume that the characteristic of $k$ is not ``bad'' for $W$ and let $e$ be the smallest $i \geq 2$ such that $1+q+q^2+... +q^{i-1}=0$. We show that the number of simple $H_k(H,q)$-modules is ``generic'', i.e., it only depends on $e$. The proof uses some computations in the {\sf CHEVIE} package of {\sf GAP} and known results due to Dipper--James, Ariki--Mathas, Rouquier and the author.

In this paper we construct new Beauville surfaces with group either $\PSL(2,p^e)$, or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion.

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the p-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups. We also classify the irreducible characters of quasi-simple groups of Lie type that are constant on the non-identity unipotent elements. In particular, we show that for groups of BN-pair rank greater than 2 the Steinberg and the trivial characters are the only characters in question. Additionally, we determine all irreducible characters whose degrees differ by 1 from the degree of the Steinberg character.

We show that the only finite nonabelian simple groups which admit a locally linear, homologically trivial action on a closed simply connected 4-manifold $M$ (or on a 4-manifold with trivial first homology) are the alternating groups $A_5$, $A_6$ and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups occur). The situation depends strongly on the second Betti number $b_2(M)$ of $M$ and has been known before if $b_2(M)$ is different from two, so the main new result of the paper concerns the case $b_2(M)=2$. We prove that the only simple group that occurs in this case is $A_5$, and then give a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

In this paper, we classify the finite simple groups with an abelian Sylow subgroup.

This is a collection of examples showing how the GAP system can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in the paper "Probabilistic generation of finite simple groups, II" by Thomas Breuer, Robert M. Guralnick, and William M. Kantor. The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP code shown for the examples can be used as test input for automatic checking of the data and the functions used.

A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le {\rm Aut}(\Gamma)$ is transitive on $V\Gamma$ and transitive on the set of $s$-arcs of $\Gamma$, where for an integer $s \ge 1$ an $s$-arc of $\Gamma$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $\Gamma$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $\Gamma$ is called 2-transitive if it is $({\rm Aut}(\Gamma), 2)$-arc-transitive but not $({\rm Aut}(\Gamma), 3)$-arc-transitive. A Cayley graph $\Gamma$ of a group $G$ is called normal if $G$ is normal in ${\rm Aut}(\Gamma)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $\Gamma$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $\Gamma$ is normal or $G$ is one of the groups ${\rm PSL}_2(11)$, $M_{11}$, $M_{23}$ and $A_{11}$. In the present paper we prove further that among these four groups only $M_{11}$ produces connected tetravalent 2-transitive non-normal Cayley graphs, and there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

We say that a finite almost simple $G$ with socle $S$ is admissible (with respect to the spectrum) if $G$ and $S$ have the same sets of orders of elements. Let $L$ be a finite simple linear or unitary group of dimension at least three over a field of odd characteristic. We describe admissible almost simple groups with socle $L$. Also we calculate the orders of elements of the coset $L\tau$, where $\tau$ is the inverse-transpose automorphism of $L$.

We provide new bounds for the divisibility function of the free group F_2 and construct short laws for the symmetric groups Sym(n). The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6 and S is covered by K^c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n,F), F an arbitrary field.

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain,Gamburd and Varj\'u it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for square-free moduli m which are relatively prime to some number m_0 the Cayley graphs Gamma(SL(d,m),pi_m(S)) form an expander family.

A group homomorphism eta:H-->G is called a localization of H if every homomorphism phi:H-->G can be `extended uniquely' to a homomorphism Phi:G-->G in the sense that Phi eta=phi. Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation A_n-->SO_{n-1}(R) of the alternating group A_n, which turns out to be a localization for n even and n>9. Dror Farjoun asked if there is any upper bound in cardinality for localizations of A_n. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group H, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space K(H,1) for any non abelian finite simple group H.

For a finite group generated by involutions, the involution width is defined to be the minimal $k\in\mathbb{N}$ such that any group element can be written as a product of at most $k$ involutions. We show that the involution width of every non-abelian finite simple group is at most $4$. This result is sharp, as there are families with involution width precisely 4.