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.