\titleofpresentation{Triply factorized groups and nearrings}
\presenter{Bernhard~Amberg}

In the theory of factorized  groups $G= AB$ with subgroups $A$ and
$B$ one often has to deal with groups of the following form
$G=AB=AM=BM$, where the normal subgroup $M$ of $G$ satisfies the
intersection property $A\cap M = A\cap M = 1$.  Such triply
factorized groups---even with nonabelian $M$---may be constructed
by using (local) nearrings.

A nearring $R$ is an algebraic structure with two operations ``${}+{}$''
and ``${}\cdot{}$'' such that $R$ is a not necessary abelian group under the
operation ``${}+{}$'' and $R$ is a semigroup under the operation ``${}\cdot{}$'' so
that a one-sided distributive law holds. A nearring $R$ with a
unit element $1$ is called local if the set of all invertible
elements forms a subgroup of the additive group of $R$.

If $L$ is the subgroup of all non-invertible elements of the local
nearring $R$, then the set $1+L$ is a subgroup of the
multiplicative group of $R$ acting on $L$, such that the
semidirect product of $(1+L)$ with $L$ is a triply factorized
group $G$, where $M$ is isomorphic to $L$ and $A$ and $B$ are both
isomorphic to $1+L$.

Conversely, it can be shown that for every triply factorized group
$G$ with $A\cap B = 1$  there exists a sub-nearring of the
nearring of all mappings from $M$ into itself,  such that the
given triply factorized group can be obtained via this
construction.

\presenteraddress{Mathematisches Institut der Universitaet Mainz, 
D-55099, Mainz, Germany}

\email{amberg@mathematik.uni-mainz.de}

