David Easdown (University of Sydney)
Minimal faithful permutation representations of groups
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. We sketch a proof of what appears to be the nontrivial fact that mu(G x G)= mu(G) if and only if G is trivial. The proof relies on a theorem of Wright that mu(G x H)=mu(G)+mu(H) if G x H is nilpotent. 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. (A prime p is Mersenne with respect to an integer q if p = 1+q+...+q^k for some k.)
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