New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before.

The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

McKay Quivers and Lusztig Algebras of Some Finite Groups

We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde {A}(\mathsf {G})$ A ~ ( G ) of a finite group $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) , which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.