Rex Dark’s research while affiliated with Ollscoil na Gaillimhe – University of Galway and other places

Let π$\pi$ be a set of primes. We show that π$\pi$-separable groups have a conjugacy class of F${\mathfrak {F}}$-injectors for suitable Fitting classes F${\mathfrak {F}}$, which coincide with the usual ones when specializing to soluble groups.

Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. This paper presents an extension of these facts to $\pi$ π -separable groups, for sets of primes $\pi$ π , by proving the existence of a conjugacy class of subgroups in $\pi$ π -separable groups, which specialize to Carter subgroups within the universe of soluble groups. The approach runs parallel to the extension of Hall theory from soluble to $\pi$ π -separable groups by Čunihin, regarding existence and properties of Hall subgroups.

If C is a class of groups, a C-injector of a finite group G is a subgroup V of G with the property that V∩K is a C-maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschütz and B. Hartley states the existence and conjugacy of F-injectors in finite soluble groups for Fitting classes F. We shall show that for groups of nilpotent length at most 4, F-injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.

Let $\pi$ be a set of primes. We show that $\pi$-separable groups have a conjugacy class of $\mathfrak F$-injectors for suitable Fitting classes $\mathfrak F$, which coincide with the usual ones when specializing to soluble groups.

In response to an Open Question of Doerk and Hawkes [5, IX Section 3, page 615] , we shall show that if Z π Z π is the Fitting class formed by the finite solvable groups whose π -socle is central (where π is a set of prime numbers), then the Z π Z π -injectors of a finite solvable group G permute with the members of a Sylow basis in G . The proof depends on the properties of certain extraspecial groups [4] .

In response to an Open Question of Doerk and Hawkes (1992) [2, IX §4, p. 628], we shall describe three constructions for the ZπZπ-injectors of a finite solvable group, where ZπZπ is the Fitting class formed by the finite solvable groups whose π-socle is central (and π is a set of prime numbers).

Suppose G is a finite solvable group, and H is a subgroup with a normal complement in G. We shall find necessary and sufficient conditions (some of which are related to the properties of coprime actions) for H to be an injector in G. We shall also use these criteria to find characterizations of injectors which need not have a normal complement.

Crystals of benzoic acid grown from dichloromethane and acetonitrile are greatly extended along the b-axis. The enhanced growth is not related to hydrogen bonding as benzoic acid is respectively dimeric and monomeric in these solvents. A mechanism is suggested for enhanced growth in the π-stacking direction for flat π-stacking systems. Facilities have been added to the Oscail software package which provide attachment energy calculations, crystal surface analysis and crystal visualization. Crystal surface analysis can be used to find the π-stacking direction and to identify the density of available hydrogen bond donors and acceptors on crystal faces. Observed and calculated morphologies for crystals of sulfathiazole form 2 grown from ethanol are in good agreement. Differences in the observed and calculated shapes of sulfathiazole forms 1, 3, 4 and 5 are attributed to solvent effects which correlate with the density of available hydrogen bond acceptors on crystal faces.

In response to a question of Doerk and Hawkes [Finite soluble groups, de Gruyter, 1992, p. 553], we shall obtain characterizations of the injectors of a finite solvable group (without recourse to the concept of a Fitting set), and we also answer in the negative a question in [Dark and Feldman, J. Group Theory 9: 2006, p. 785].