Mathematical Research Seminar - Archive

We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups.  This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\Z_{p_1}^{a_1}\times\cdots\times\Z_{p_r}^{a_r}$, $a_i\le 4$ are isomorphic if and only if they are isomorphic by a group automorphism of $\Z_{p_1}^{a_1}\times\cdots\times\Z_{p_r}^{a_r}$.  That is, that such groups are CI-groups with respect to digraphs.  Finally, we will discuss (perhaps tractable) open problems and conjectures.

It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not admit even realizations with points and pseudolines, i.e. they are 
not topological. In this paper we show that every combinatorial configuration can be realized as a quasiline arrangement on a real projective plane. A quasiline arrangement can be viewed as a map on a closed surface. Such a map can be used to distinguish between two "distinct" realizations of a combinatorial configuration as a quasiline arrangement. Based on work in progress with several mathematicians including Leah Berman, Juergen Bokowski, Gabor Gevay, Jurij Kovič and Arjana Žitnik.