Classification Of Finite Simple Groups Research Articles

Finite groups whose first Cartan invariants over a field of characteristic two are two

We determine all finite groups whose first Cartan invariants over a field of characteristic two are two, by using the classification of finite simple groups.

Boosting an analogue of Jordan's theorem for finite groups

Let C be a set of finite groups which is closed under taking subgroups and let d and M be positive integers. Suppose that for every G∈C whose order is divisible by at most two distinct primes there exists an abelian subgroup A⊆G such that A is generated by d or fewer elements and [G:A]≤M. We prove that there exists a positive constant C0 such that every G∈C has an abelian subgroup A satisfying [G:A]≤C0, and A can be generated by d or fewer elements. We also prove some related results. Our proofs use the Classification of Finite Simple Groups.

S-Conditionally Permutable Subgroups and p-Nilpotency of Finite Groups

We study the p-nilpotency of a group such that every maximal subgroup of its Sylow p-subgroups is s-conditionally permutable for some prime p. By using the classification of finite simple groups, we get interesting new results and generalize some earlier results.

A dual version of Huppert's conjecture on conjugacy class sizes

Abstract In [J. Algebra 344 (2011), 205–228], a conjecture of J. G. Thompson for PSL n (q) was proved. It was shown that every finite group G with the property Z(G) = 1 and cs(G) = cs(PSL n (q)) is isomorphic to PSL n (q) where cs(G) is the set of conjugacy class sizes of G. In this article we improve this result for PSL2(q). In fact we prove that if cs(G) = cs(PSL2(q)), for q > 3, then G ≅ PSL2(q) × A, where A is abelian. Our proof does not depend on the classification of finite simple groups.

The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k G ) of the trivial kG-module k G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.

Representation of finite groups: conjectures, reductions, and applications

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

2-blocks with abelian defect groups

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovanʼs conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form C2m×C2m for m⩾2 has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovanʼs conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.

ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

Tetravalent edge-transitive graphs of order p 2 q

A graph is called edge-transitive if its full automorphism group acts transitively on its edge set. In this paper, by using classification of finite simple groups, we classify tetravalent edge-transitive graphs of order p2q with p, q distinct odd primes. The result generalizes certain previous results. In particular, it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.

Finite factorizable groups with solvable ℙ2-subnormal subgroups

A subgroup H of a finite group G is called ℙ2-subnormal whenever there exists a subgroup chain H = H0 ≤ H1 ≤ ... ≤ Hn = G such that |Hi+1: Hi| divides prime squares for all i. We study a finite group G = AB on assuming that A and B are solvable subgroups and the indices of subgroups in the chains joining A and B with the group divide prime squares. In particular, we prove that a group of this type is solvable without using the classification of finite simple groups.

Quasisimple classical groups and their complex group algebras

Let H be a finite quasisimple classical group, i.e., H is perfect and S:= H/Z(H) is a finite simple classical group. We prove that, excluding the open cases when S has a very exceptional Schur multiplier such as PSL3(4) or PSU4(3), H is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.

OD-Characterization of the Projective Special Linear Groups L2(q)

Let L2(q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L2(q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi.

2-Blocks with minimal nonabelian defect groups II

Abstract. We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.

Problems of Connectivity between the Sylow Graph,the Prime Graph and the Non-Commuting Graph of a Group

The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid CFSG or not. On the other hand, what happens for infinite groups? Since the status of knowledge of the non-commuting graph and of the prime graph is satisfactory, is it possible to find relations between these two graphs and the Sylow graph? In the present note we make the point of the situation and formulate the above questions in appropriate way.

The Applications of Group Theory

Since the classification of finite simple groups completed last century, the applications of group theory are more and more widely. We first introduce the connection of groups and symmetry. And then we respectively introduce the applications of group theory in polynomial equation, algebraic topology, algebraic geometry , cryptography, algebraic number theory, physics and chemistry.

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Using the classification of finite simple groups, we prove Alperin's weight conjecture and the character theoretic version of Broué's abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification

Davenport’s Problem asks: What can we expect of two polynomials, over Z, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, Lewis and Schinzel. By bounding the degrees, but expanding the maps and variables in Davenport’s Problem, Galois stratification enhanced the separated variable theme, solving an Ax and Kochen problem from their Artin Conjecture work. Denef and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement. By restricting the variables, but leaving the degrees unbounded, we found the striking distinction between Davenport’s problem over Q, solved by applying the Branch Cycle Lemma, and its generalization over any number field, solved by using the simple group classification. This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups. Guralnick and Thompson led its solution in stages. We look at two developments since the solution of Davenport’s problem. We use these to consider: Going beyond the simple group classification to handle imprimitive groups, and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

On a relation between the Sylow and Baer-Suzuki theorems

Given a set π of primes, say that a finite group G satisfies the Sylow π-theorem if every two maximal π-subgroups of G are conjugate; equivalently, the full analog of the Sylow theorem holds for π-subgroups. Say also that a finite group G satisfies the Baer-Suzuki π-theorem if every conjugacy class of G every pair of whose elements generate a π-subgroup itself generates a π-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow π-theorem then it satisfies the Baer-Suzuki π-theorem as well.

On endo-trivial modules for p-solvable groups

In this note, we will prove a conjecture of Carlson et al. [1], thus resolving the question left open in [1] and completing the classification of torsion endo-trivial modules for p-solvable groups. Namely, we will prove that if G is a finite p-nilpotent group which contains a non-cyclic elementary Abelian p-subgroup and k is an algebraically closed field of characteristic p, then all simple endo-trivial kG-modules are 1-dimensional. In fact, we do rather more: we prove the analogous result directly in the case that G is p-solvable and contains an elementary Abelian p-subgroup of order p2. Carlson, Mazza and Thevenaz had reduced the proof of this result for p-solvable G to the p-nilpotent case (and had proved the result in the solvable case), but our method is somewhat different. Our proof does require the classification of finite simple groups. Specifically, we require the well-known fact that the outer automorphism group of a finite simple group of order prime to p has cyclic Sylow p-subgroups (see, for example, Theorem 7.1.2 of Gorenstein et al. [4]). Let us recall that a kG-module M is endo-trivial if M ⊗ M∗ ∼= k ⊕ N , where N is a projective kG-module. If |G| is divisible by p, then any endo-trivial kG-module has dimension prime to p.The vertex of any indecomposable endo-trivial kG-module is a Sylow p-subgroup of G. We remark that if M is an endo-trivial kG-module which is not 1-dimensional, then a Sylow p-subgroup of G acts faithfully on M ⊗ M∗ and hence acts faithfully on M.

Denseness and Zariski denseness of Jones braid representations

Using various tools from representation theory and group theory, but without using hard classification theorems such as the classification of finite simple groups, we show that the Jones representations of braid groups are dense in the (complex) Zariski topology when the parameter t is not a root of unity. As first established by Freedman, Larsen and Wang, we obtain the same result when t is a nonlattice root of unity, other than one initial case when t has order 10. We also compute the real Zariski closure of these representations (meaning, the closure in Zariski closure of the real Weil restriction). When such a representation is indiscrete in the analytic topology, then its analytic closure is the same as its real Zariski closure. In this article we will study representations of braid groups associated with the Jones polynomial J.L;t/. Our question is to determine the closures of these representations, which are then Lie groups. Freedman, Larsen and Wang [4] computed these closures in the important case where tD exp.2 i=r/ is a principal root of unity. In this case, the (reduced) braid representations are unitary, and the braids can be interpreted as quantum circuits. Freedman, Larsen and Wang established that the Jones representations are eventually dense if rD 5 or r 7. This has the important corollary that these representations are universal for quantum computation. In this article t will usually be a complex number which is not a root of unity. Although the Jones polynomial is not directly a model of quantum computation for these values of t , the closure of the braid group representation is still interesting for related questions in complexity theory; see Ahargonov, Arad, Eban and Landau [1]. Also, we will say more about the Zariski closure of the braid group action in the target group GL.N;C/, rather than the closure in the usual topology. Switching to the Zariski topology simplifies the question, and yet in many cases it does not change the question very much. To be precise, we will consider two Zariski topologies on GL.N;C/ and SL.N;C/. The usual Zariski topology is generated by polynomials in the complex matrix entries, and