Colorings of hypermaps and a conjecture of Brenner and Lyndon (Q788000)

Following \textit{R. Cori} [Astérisque 27, 1-169 (1975; Zbl 0313.05115)], a hypermap is defined combinatorially as a pair of permutations (\(\alpha\),\(\beta)\) on an n-set \(\Omega\) such that the group generated by \(\alpha\) and \(\beta\) is transitive on \(\Omega\). The cells of the hypermap are of orbits of \(\alpha\) (the vertices), \(\beta\) (the edges), and \(\alpha\beta\) (the faces). An automorphism of (\(\alpha\),\(\beta)\) is a permutation of \(\Omega\) which commutes with both \(\alpha\) and \(\beta\). The main theorem of this paper states that if \(\alpha\beta\) is transitive on \(\Omega\) and \(\alpha^ p=\beta^ q=1\), where p and q are distinct primes, then the automorphism group of (\(\alpha\),\(\beta)\) is either trivial or a cyclic group of order p,q or pq. This problem was posed by \textit{J. L. Brenner} and \textit{R. C. Lyndon} [Pac. J. Math. 104, 285-315 (1983; Zbl 0507.05056)] in the special case when \(p=3\) and \(q=2\). To prove the main result a theory of colorings of hypermaps is developed and an auxiliary theorem of independent interest is proved: a hypermap automorphism of prime order cannot fix exactly one cell.