The study of transitive permutation groups of prime degree can be traced back two and a half centuries, through Burnside and Galois, to the work of Lagrange on polynomials of prime degree. It is sometimes asserted that the groups of prime degree are now completely known, as a consequence of the classification of finite simple groups: apart from a few interesting but easily-understood exceptions, there are infinite families of affine, alternating and symmetric groups, together with various projective groups related to $PSL_n(q)$, all acting naturally in those cases when their natural degree is prime. Although true, this assertion ignores an apparently difficult number-theoretic problem, namely whether or not there exist infinitely many primes equal to the natural degree $(q^n-1)/(q-1)$ of $PSL_n(q)$. Such primes are also relevant to alternative versions of Waring's problem. In joint work with Sasha Zvonkin I shall present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one considers only prime values of $q$.

Ural Seminar on Group Theory and Combinatorics