Green's correspondence of group modules finds its simplest expression when a finite multiplicative group $G$ has a trivial intersection Sylow $p$-subgroup $P$, for some prime $p$. Then it is between all isomorphism classes of projective-free $\mathbf{R}G$-lattices $\mathbf{L}$ and all isomorphism classes of projective-free $\mathbf{R}N$-lattices $\mathbf{K}$, where $\mathbf{R}$ is a suitable valuation ring and $N$ is the normalizer of $P$ in $G$. In that case we show in Theorem 3.2 below that the $\mathbf{R}G$-lattice $\mathbf{L}$ is determined by its associated lattices over the residue field and field of fractions of $\mathbf{R}$ if and only if $\mathbf{K}$ has this same property. By Theorem 3.7 some important $\mathbf{R}G$-lattices $\mathbf{L}$ have this property of being “rationally determined.” So it would be worthwhile to see if the $\mathbf{R}N$-lattices with this property (and perhaps with other properties preserved by this Green correspondence) could be classified.

Let $\mathcal{G}$ be a geometry in which the elements of one type are called points and the elements of some other type are called lines. Suppose that every line is incident to exactly $p + 1$ points where $p$ is a prime number. A (non-abelian) representation of $\mathcal{G}$ is a pair $(R, \psi)$, where $R$ is a group and $\psi$ is a mapping of the set of points of $\mathcal{G}$ into the set of subgroups of order $p$ in $R$ such that $R$ is generated by the image of $\psi$ and whenever $\{x_{\infty}, x_0, \dots, x_{p-1}\}$ is the set of points incident to a line, the subgroups $\psi (x_{\infty}), \psi (x_0), \dots, \psi (x_{p-1})$ are pairwise different and generate in $R$ a subgroup of order $p^2$. In this article we discuss representations of some classical and sporadic geometries and their applications to certain problems in algebraic combinatorics and group theory.

We will consider some vertex operator algebras (VOAs) whose automorphism groups are generated by 3-transpositions. Our main examples are some code VOAs. We will classify the structures of the automorphism groups of the code VOAs. We give explicit constructions of such code VOAs, and determine the full automorphism groups for some cases.

There is a long running project due to U. Meierfrankenfeld and the author to investigate the so called small modules for the finite simple groups. These modules show up in the amalgam method which recently became important for the revision of parts of the classification of the finite simple groups. A small module either is a quadratic module or a module on which an elementary abelian group acts such that the codimension of the centralizer is small compared with its order. In this paper we determine all irreducible modules $V$ over $GF(2)$ for the finite simple groups $G$ such that $|V : C_V(A)| \le |A|^2$ for some nontrivial elementary abelian subgroup $A$ of $G$ where in addition we have $[V, A, A] = 1$.

For the sporadic Suzuki simple group, the radical $p$-subgroups for $p = 2$ and 3 are classified and the simplicial complex of their chains is shown to be homotopically equivalent to a $p$-local geometry. Further investigation of the related complexes for $p = 2$ gives a counterexample to Conjecture 1 in [4].