Classification of regular maps of negative prime Euler characteristic

Abstract

Abstract. We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus p + 2 where p is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus p + 2 where p is a prime such that p ≡ 1 (mod 12) and p = 13.

A computer-free classification of orientably-regular maps on surfaces of genus p+1 for prime p

“…(a) M has type (8,8) and G ∼ = C p ⋊ C 8 , of order 8p, with p ≡ 1 mod 8, (b) M has type (5,10) and G ∼ = C p ⋊ C 10 , of order 10p, with p ≡ 1 mod 10, (c) M has type (6,6) and G ∼ = C p ⋊ (C 6 × C 2 ), of order 12p, with p ≡ 1 mod 6.…”

“…Now these include the eight pairs (3,15), (3,24), (3,42), (4,20), (6,12), (8,24), (9,18) and (12,12), with |G| = 2µ(k, ℓ)p = 20p, 16p, 14p, 10p, 8p, 6p, 6p and 6p, respectively, but as p ≥ 5, none of the listed group orders is divisible by the first entry k of the corresponding pair (k, ℓ), contradicting the fact that one generator of G has order k. Hence we can eliminate all of them from consideration. Also we can eliminate two further pairs, namely (7,42) and (5,20): in the former case, |G| = 2µ(k, ℓ)p = 6p and is divisible by ℓ = 42, and so p = 7, but then G ∼ = C 42 which is clearly not (7, 42, 2)-generated, while in the latter case |G| = 2µ(k, ℓ)p = 8p and is divisible by ℓ = 20, so p = 5, and then |G| = 40 and so G has a normal Sylow 5-subgroup P , but then the quotient G/P has order 8 and so is clearly not (1, 4, 2)-generated.…”

“…To complete this section, we deal with the five cases listed in Table 1 but not covered by Proposition 4.2, namely the pairs (3, 10), (4, 5), (5,5), (5,10) and (10, 15), all with p = 5. We do this by using a trivial but useful consequence of Sylow theory coupled with the fact that if a Sylow p-subgroup is normal in the commutator subgroup G ′ of G, then it is also characteristic in G ′ and therefore normal in G as well:…”

“…(For example, for λ = 8 one may take p ≥ 17.) In the same year, Breda, Nedela and Širáň [5] derived a classification of regular maps of non-orientable genus p + 2 where p is prime. While the work for [2] was based on the theory of Riemann surfaces, the approach of [5] was purely group-theoretic.…”

“…In the same year, Breda, Nedela and Širáň [5] derived a classification of regular maps of non-orientable genus p + 2 where p is prime. While the work for [2] was based on the theory of Riemann surfaces, the approach of [5] was purely group-theoretic. Methods of [5] were then substantially extended by Conder, Širáň and Tucker [12], to obtain a classification of orientably-regular maps M of genus p + 1 for any prime p ≥ 17, and hence for any prime p (thanks to earlier computer-assisted findings).…”