DEPARTMENT OF MATHEMATICS
               THE UNIVERSITY OF QUEENSLAND

               ALGEBRA/COMBINATORICS SEMINAR

           11am Tuesday 27th September, room 67-343


          Conjugacy closedness and combinatorics
                          by

                      Ales Drapal
             Charles University, Prague


ABSTRACT:

A loop is a binary system with a unit, in which equations ax=b and
ya=b have unique solutions for all a and b. Associative loops are groups.
With every element a one can associate a left translation x -> ax,
and a right translation x -> xa. A loop is said to be left conjugacy
closed (LCC) if left translations are closed for conjugation. A loop
that is both LCC and RCC is called conjugacy closed (CC). CC-loops
can be obtained from symmetric trilinear forms, which connects them
to doubly even binary codes and to linear codes over odd order fields.
The CC-loops obtained in this ways can be used as building blocks
for large local subgroups of the Monster group. Another connection
to combinatorics is through so called live objects, i.e. objects
that reproduce themselves when regarded as mappings in an approriate
way. Live 1-factorization of a graph and live Kirkman triples will
be mentioned in this context. If time allows, some recent structural
results about LCC and CC loops will be cited.


All welcome