Dihedral Sylow 2-subgroups Research Articles

Classification of regular maps of prime characteristic revisited: Avoiding the Gorenstein-Walter theorem

Breda, Nedela and Širáň (2005) classified the regular maps on surfaces of Euler characteristic −p for every prime p. This classification relies on three key theorems, each proved using the highly non-trivial characterisation of finite groups with dihedral Sylow 2-subgroups, due to D. Gorenstein and J.H. Walter (1965). Here we give new proofs of those three facts (and hence the entire classification) using somewhat more elementary group theory, using without referring to the Gorenstein-Walter theorem.

Splendid Morita equivalences for principal blocks with generalised quaternion defect groups

We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow 2-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow 2-subgroups realised by Scott modules.

Bi-rotary Maps of Negative Prime Characteristic

Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the 1970s to describe non-orientable maps with the property that opposite orientations can consistently be assigned to adjacent vertices. In contrast to orientability, which is both a combinatorial and topological property, bi-orientability is only a combinatorial property. In this paper we classify the bi-orientable maps whose local-orientation-preserving automorphism groups act regularly on arcs, called here bi-rotary maps, of negative prime Euler characteristic. Unlike other classification results for highly symmetric maps on such surfaces, we do not use the Gorenstein–Walter result on the structure of groups with dihedral Sylow 2-subgroups.

On finite simple groups acting on integer and mod 2 homology 3-spheres

We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z 2 -homology 3-sphere are the linear fractional groups PSL ( 2 , q ) , for an odd prime power q (and the dodecahedral group A 5 ≅ PSL ( 2 , 5 ) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL ( 2 , q ) acts.

The mod 2 cohomology algebras of finite groups with semidihedral sylow 2-subgroups

The mod2 cohomology algebras of finite groups with dihedral Sylow 2-subgroups are completely determined, by using the theory of relatively projective covers.

The mod 2 cohomology algebras of finite groups with dihedral sylow 2-subgroups

The mod 2 cohomology algebras of finite groups with dihedral sylow 2-subgroups

A dimension formula for the cohomology rings of finite groups with dihedral sylow 2-subgroups

A dimension formula for the cohomology rings of finite groups with dihedral sylow 2-subgroups

Characters of finite groups with dihedral Sylow 2-subgroups

In this article, we give alternate proofs of some intermediate results in the Gorenstein-Walter classification [6] of groups with dihedral Sylow 2subgroups. These intermediate results will be applied in an alternate proof [ 1 ] of the entire classification by the first author. Our proofs rely only on elementary character theory and do not use block theory, but they are motivated by the related work of Brauer [2] and of Gorenstein and Walter [6, Propositions l-41, which was obtained by block theory. Now we describe the situation which will be assumed in the following sections and in the three theorems stated below. For elementary properties of groups with dihedral Sylow 2-subgroups the reader is referred to [S, Section 7.71.

Finite groups with dihedral Sylow 2-subgroups

It is mainly based on characterizations of P&Y,(q) and A, by Brauer et al. [4] and by Suzuki [ 141, on the Feit-Thompson Odd Order Theorem [5], used throughout this paper without reference, and on Glauberman’s ZJTheorem [6], not available when Gorenstein and Walter proved their fundamental classification theorem. Our maximal subgroup approach is adopted from [ 11. It was Glauberman’s work [ 71, privately communicated, which revived the author’s interest in the subject. Via [3] it led to extensive simplifications of his original proof. In the remainder of this section we list some elementary general facts. In addition to standard notation we write F,(X) for 0,(&Y)) and 1*(s) for (xEX]xS=x-‘J. A group E is called quasisimple if 1 # E = E’ and E/Z(E) is simple.

Principal blocks of groups with dihedral sylow 2-subgroups

(1977). Principal blocks of groups with dihedral sylow 2-subgroups. Communications in Algebra: Vol. 5, No. 7, pp. 665-694.

The principal block of finite groups with dihedral sylow 2-subgroups

The principal block of finite groups with dihedral sylow 2-subgroups

The characterization of finite groups with dihedral Sylow 2-subgroups. III

This is the third of three parts. A description of the contents may be found in the introduction given in the first part. The first part was published in this journal, volume 2, pp. 85–151 (1965). The second part was published in this journal, volume 2, pp. 218–270 (1965).

The characterization of finite groups with dihedral Sylow 2-subgroups—II

This is the second of three parts. A description of the contents may be found in the introduction given in the first part which was published in this journal, volume 2, no. 1, pp. 85–151 (April 1965).

The characterization of finite groups with dihedral Sylow 2-subgroups. I

Abstract This is the second of three parts. A description of the contents may be found in the introduction given in the first part which was published in this journal, volume 2, no. 1, pp. 85–151 (April 1965).

On the maximal subgroups of finite simple groups

In the course of their proof of the solvability of groups of odd order, W. Feit and J. G. Thompson [I] establish many deep properties of the maximal subgroups of a minimal simple group 8 of odd order. Perhaps the most important of these results is the following: if for some prime p, 6 possesses an elementary abelian subgroup of order p3, then there exists a unique maximal subgroup 9X of 6 containing a given Sylow p-subgroup $3 of 6; furthermore if si is any proper subgroup of 8 such that Rn $3 possesses an elementary abelian subgroup of order p3, then R is necessarily a subgroup of 92. In our study of finite groups 6 with dihedral Sylow 2-subgroups, we require the identical result for certain odd primes p dividing the order of the centralizer of an involution in 6. On the basis of this theorem, we can show that the unique maximal subgroup 911 of 6 containing a given Sylow p-subgroup actually contains the entire centralizer of some involution in 6 and has no normal subgroups of index 2. We are then able to apply to both ?JJI and 8 formulas for the order of an arbitrary group with dihedral Sylow 2-subgroup having no normal subgroups of index 2, as developed by means of character

On finite groups with dihedral Sylow 2-subgroups

On finite groups with dihedral Sylow 2-subgroups