Le groupe $H_8\times C_2$ est le plus petit groupe pour lequel existent des modules stablement libres non libres. On montre que toutes les classes d’isomorphisme de tels modules peuvent être représentées une infinité de fois par des anneaux d’entiers. On applique un travail de classification de Swan, pour cela on doit construire explicitement des bases normales d’entiers d’extensions à groupe $H_8$; cela se fait en liant un critère de Martinet avec une construction de Witt.

Let $U$ be the two-parameter quantized enveloping algebra $U_{r,s}\left({\mathrm{𝔰𝔩}}_2\right)$ and $F\left(U\right)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F\left(U\right)$ in the case when $r{s}^{-1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group ${\mathbf{SL}}_2\left({\mathbf{F}}_{41}\right)\otimes {\tilde{S}}_3\subset {\mathbf{GL}}_{80}\left(\mathbf{Z}\right)$ is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the $\Theta$-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product...

Given a groupoid $\langle G,☆\rangle$, and $k\ge 3$, we say that $G$ is antiassociative if an only if for all $x_1,x_2,x_3\in G$, $(x_1☆x_2)☆x_3$ and $x_1☆(x_2☆x_3)$ are never equal. Generalizing this, $\langle G,☆\rangle$ is $k$-antiassociative if and only if for all $x_1,x_2,...,x_k\in G$, any two distinct expressions made by putting parentheses in $x_1☆x_2☆x_3☆\cdots ☆x_k$ are never equal. We prove that for every $k\ge 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

It is well known that given a Steiner triple system one can define a quasigroup operation $·$ upon its base set by assigning $x·x=x$ for all $x$ and $x·y=z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems....