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Strong map-symmetry of $SL (3, K)$ $SL (3, K)$ and $PSL (3, K)$ $PSL (3, K)$ for every finite field $K$ $K$

Antonio Breda d’Azevedo

Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

E-mail Address: breda@ua.pt

Corresponding author.

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and

Domenico A. Catalano

https://orcid.org/0000-0002-1542-9614

Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

E-mail Address: domenico@ua.pt

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https://doi.org/10.1142/S0219498821500481Cited by:1 (Source: Crossref)

Abstract

In this paper, we show that for any finite field $K$ $K$ , any pair of map-generators (that is when one of the generators is an involution) of $SL (3, K)$ $SL (3, K)$ and $PSL (3, K)$ $PSL (3, K)$ has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group $SL (3, K)$ $SL (3, K)$ or $PSL (3, K)$ $PSL (3, K)$ is reflexible, or equivalently, there are no chiral regular maps with automorphism group $SL (3, K)$ $SL (3, K)$ or $PSL (3, K)$ $PSL (3, K)$ . As remarked by Leemans and Liebeck, also $SU (3, K)$ $SU (3, K)$ and $PSU (3, K)$ $PSU (3, K)$ are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

Communicated by M. L. Lewis

Keywords:

AMSC: Primary: 20E45, Primary: 20B25, Primary: 20F05, Primary: 20G40, Secondary: 05E18, Secondary: 57M15

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