Intrinsic Geometries of the Simple Group of Order 168
type: Item
instance of: scholarly article
title: Intrinsic Geometries of the Simple Group of Order 168
DOI: 10.1080/00029890.2024.2368589
author: Ian Stewart
published in: American Mathematical Monthly
publication date: 2024-12-16T00:00:00Z
review text: Among the finite groups of order \(168\), there is, up to isomorphism, a unique simple group \(G\). Interestingly, it has a twofold nature, as \(\mathrm{PSL}(2,7) \simeq G \simeq \mathrm{PSL}(3,2)\). In this paper the author show that \(G\) has a natural action on the projective line over \(\mathbb{F}_{7}\), so it is isomorphic to \(\mathrm{PSL}(2,7)\), and a natural action on the projective plane over \(\mathbb{F}_{2}\), so it is isomorphic to \(\mathrm{PSL}(3,2)\). The proof relies solely on elementary methods of group theory.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 7958665
MaRDI profile type: Publication
cites work: Sui postlilati fondamentali della geometria projettiva in uno spazio lineare a un numero qualunque di dimensioni.
cites work: Linear groups with an exposition of the \textit{Galois} field theory.
cites work: A proof by graphs that PSL(2,7)\(\cong PSL(3,2)\)
cites work: On the seventh order transformation of elliptic functions.
cites work: The Klein quartic in number theory
cites work: Why is PSL(2,7) ≅ GL(3,2)?
cites work: Q4423053
cites work: Q4488161
cites work: The Strong Law of Small Numbers
cites work: Q4877538
cites work: Q4282627
cites work: Q5331572
cites work: Q3684278
description: scientific article; zbMATH DE number 7958665