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The Global Cohen-Lenstra Heuristic
The Cohen-Lenstra heuristic is a universal principle that assigns to each
group a probability that tells how often this group should occur "in nature".
The most important, but not the only, applications are sequences of class
groups, which behave like random sequences of groups with respect to the
so-called Cohen-Lenstra probability measure.
So far, it was only possible to define this probability measure for finite
abelian -groups. We prove that it is also possible to define an analogous
probability measure on the set of \emph{all} finite abelian groups when
restricting to the -algebra on the set of all finite abelian groups
that is generated by uniform properties, thereby solving a problem that was
open since 1984
Regular dessins with a given automorphism group
Dessins d'enfants are combinatorial structures on compact Riemann surfaces
defined over algebraic number fields, and regular dessins are the most
symmetric of them. If G is a finite group, there are only finitely many regular
dessins with automorphism group G. It is shown how to enumerate them, how to
represent them all as quotients of a single regular dessin U(G), and how
certain hypermap operations act on them. For example, if G is a cyclic group of
order n then U(G) is a map on the Fermat curve of degree n and genus
(n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus
274218830047232000000000000000001. For other non-abelian finite simple groups,
the genus is much larger.Comment: 19 page
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