Strongly Real Special 2-Groups

Strongly real groups and totally orthogonal groups form two important subclasses of real groups. In this article we give a characterization of strongly real special 2-groups. This characterization is in terms of quadratic maps over fields of characteristic 2. We then provide examples of groups which are in one subclass and not the other. It is a conjecture of Tiep that such examples are not possible for finite simple groups.     

On Weyl groups in minimal simple groups of finite Morley rank

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type.     

Shelling the coset poset

It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen-Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups.     

Chermak–Delgado simple groups

This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvableChermak–Delgado simple groups, which the author defines. The notion of “Chermak–Delgado simple,” or “CD-simple” for short, is a generalization of simple groups through the Chermak–Delgado lattice. The author completes a classification of Chermak–Delgado simple groups under certain restrictions on the primes involved in the group order.     

用元素阶的集合和Sylow子群的正规化子阶的集合刻画奇数维的有限正交单群

本文研究了一类单群的数量刻画问题.运用有限单群分类定理及群的元素阶的素图表示的结构定理,获得了有限群G同构于有限正交单群PΩ2n+1(q)(n≥13),当且仅当群的元素的阶的集合一致和群的Sylow子群的正规化子的阶之集合一致.在某种意义推进施武杰教授的一个猜想.     

Carter subgroups of finite groups

It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained.     

Finite groups with a five-component prime graph

Some classification is obtained for the finite groups whose prime graph has five connected components. In particular, we show that such a group is simple. Some related results are stated about representations of simple groups.     

Complete Classification of BKM Lie Superalgebras Possessing Special Imaginary Roots

In this article, we give a complete classification of BKM Lie superalgebras possessing special imaginary roots. In this classification, we have also included infinite dimensional BKM Lie superalgebras with finite or infinite number of real or imaginary simple roots.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

Finite Groups with Nonmonomial Characters of Prime Degree

It is proved that every finite group for which the degrees of its nonmonomial characters are primes is solvable. The proof uses the classification of the finite simple groups.     

Finite groups with Hall supplements to primitive subgroups

Using the classification of finite simple groups, we establish the solvability of every finite group with solvable Hall supplements to primitive subgroups. We prove also that if those supplements are Ore dispersive then so is the group itself.     

Standard components of alternating type centralized by a 4-group

We determine the finite groups possessing a standard subgroup which is the covering group of an alternating group, and whose centralizer is of 2-rank at least 2. The result is one step in the original proof of the classification of the finite simple groups, and presumably also in improvements to that proof. The paper is a modification of the original preprint, which had remained unpublished since the seventies.     

Group factorisations,uniform automorphisms,and permutation groups of simple diagonal type

We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.     

Distance Transitive Generalized Quadrangles of Prime Order

Without using the classification of finite simple groups, we classify the finite generalized quadrangles of prime order admitting a group acting distance transitively on the collinearity graph. Our method uses combinatorial geometry and permutation groups.     

Strongly closed subgroups of finite groups

This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.     

On the Automorphism Groups of Affine-Invariant Codes.

Using the classification of finite simple groups, we obtain a condition for a permutation group on a finite field GF(pm) to have the affine group AGL(1,pm) as a subgroup. Applying this result to the automorphism groups of non trivial affine-invariant codes, we prove that these automorphism groups are always subgroups of the general affine group AGL(m,p).     

Classification of maximal subgroups of odd index in finite simple classical groups

The classification of maximal subgroups of odd index in finite simple classical groups is obtained.     

Suzuki群与Steiner 4设计

本文研究了非平凡Steiner 4设计的自同构群是旗传递的情形.利用有限2传递置换群的分类,得到了旗传递非平凡Steiner 4设计的自同构群的基柱不是Suzuki群.     

On a class of butler groups

A complete classification, both up to quasi isomorphism and up to isomorphism, is given of the strongly indecomposable torsionfree abelian groups that occur as quotients of a finite rank completely decomposable torsionfree group modulo a rank 1 subgroup. This research was done while the second author held a visiting professorship at Tulane University.