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Identifying long cycles in finite alternating and symmetric groups acting on subsets
Let be a permutation group on a set , which is permutationally
isomorphic to a finite alternating or symmetric group or acting on
the -element subsets of points from , for some arbitrary but
fixed . Suppose moreover that no isomorphism with this action is known. We
show that key elements of needed to construct such an isomorphism
, such as those whose image under is an -cycle or
-cycle, can be recognised with high probability by the lengths of just
four of their cycles in .Comment: 45 page
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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