Algebra Seminar: Easdown -- Minimal Faithful Permutation Representations of Groups with Focus on Pathological Behaviour with Respect to Direct Products

David Easdown (The University of Sydney) 

Friday 26 Feb, 12-1pm, Place: New Law School Lecture Theatre 026 

Minimal Faithful Permutation Representations of Groups with Focus on Pathological
Behaviour with Respect to Direct Products.  

The minimal faithful degree mu(G) of a finite group G is the least nonnegative integer n
such that G embeds in the symmetric group on n letters.  Finding mu(G) is equivalent to
solving a minimisation problem in the lattice of subgroups of G.  Interesting and rich
connections come into play, intertwining the underlying number-theoretic relationships
between subgroups and the sublattice of normal subgroups.  Apparently difficult
fundamental questions arise about the behaviour of mu with respect to taking direct
products and homomorphic images.  Pathological behaviour is especially interesting from
our point of view, and we focus on the property, when it occurs, of a group G absorbing
a group H as a direct factor in the sense that mu(G x H) = mu(G).  We sketch a proof of
what appears to be the nontrivial fact that mu(G^n x G)= mu(G^n) if and only if G is
trivial, so that it is impossible for a direct power of a nontrivial group to absorb a
copy of itself.  The proof relies on a theorem of Wright that mu(G x H)=mu(G)+mu(H) if G
x H is nilpotent and classical techniques originating with Zassenhaus and questions
about minimal regular actions and embeddings in wreath products.  We also construct a
finite group G that does not decompose nontrivially as a direct product, but such that
mu(G x H) = mu(G) for an arbitrarily large direct product H of elementary abelian groups
(with mixed primes).  A simplification of the construction depends on the existence of
infinitely many primes that do not have the Mersenne property, which itself appears to
be nontrivial, and a consequence of the Green-Tao theorem about the existence of
arbitrarily long arithmetic sequences of primes.  (A prime p is Mersenne with respect to
an integer q if p = 1+q+...+q^k for some k.)  We also construct inside Sym(40) what we
believe is the smallest example where mu(G x H) = mu(G) and H is nonabelian.