Regular Maps Constructed from Linear Groups

✍ Scribed by Peter McMullen; Barry Monson; Asia Ivić Weiss

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 380 KB
• Volume: 14
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1993.1057

✦ Synopsis

For an integer (m>2), let (L_{2}^{\langle-1\rangle}\left(\mathbb{Z}{m}\right)) be the group of (2 \times 2) matrices over (\mathbb{Z}{m}) with determinants (\pm 1). Then for each subgroup (H) with ({ \pm 1} \leqslant H \leqslant \operatorname{centre}\left(L_{2}^{(-1)}\left(\mathbb{Z}{m}\right)\right)) there exists a regular map of type ({3, m}) whose group is (L{2}^{(-1)}\left(\mathbb{Z}{m}\right) / H). Ways are determined in which one such map can cover another, and in which a map decomposes as a blend of simpler maps. In all this, the prime factorization of (m) plays the key role. In some cases the structural results enable a simple description of the corresponding automorphism groups. Also described are realizations for the maps based on the 'plane' (\mathbb{Z}{m}^{2}).