Finite groups of 2 X 2 integer matrices

• Main Authors: Castro, Maria Katrina, Lu, John Ronald
• Format: text
• Language: English
• Published: Animo Repository 2002
• Online Access: https://animorepository.dlsu.edu.ph/etd_bachelors/17228
• Institution: De La Salle University

This thesis is based mainly on Sections 1 to 5 of the article "Finite Groups of 2 X 2 Integer Matrices" by George Mackiw which appeared in Mathematics Magazine, Volume 69 (1996). This study was motivated by a presentation intended to show that the dihedral group D6 of symmetries of the hexagon can be realized as a group of invertible 2 x 2 matrices with real number coefficients. It discusses some of the properties of the general linear group GL(2,Z0, the set of invertible 2 x 2 integer matrices whose inverses also have integer entries and some properties of the minimum polynomial of a matrix. The Hamilton-Cayley Theorem were used to prove some of these properties. The special linear group SL(2,Z) is the subgroup of all matrices in GL(2,Z) with determinant 1. In this thesis, the order of SL(2,Z) is computed and its elements are enumerated. The main result in this thesis states that a finite group G can be represented as a group of invertible 2 x 2 integer matrices if and only if G is isomorphic to the subgroup of D4 or D6.