Automorphisms of the modular and extended modular groups

Abstract

The groups of 2 x 2 integral matrices GL(2, Z) and SL(2, Z) and their related projective groups PGL(2, Z) and PSL(2, Z) have been used and studied for over a century, but little work has been done on their automorphism groups. Hua and Reiner in the early 1950's gave generators for all these automorphism groups (not only for 2 x 2 , but also for n x n matrix groups) showing, in particular, that the groups PGL(n, Z) have only inner automorphisms. We show that in fact n = 2 is an exception, having outer automorphisms too, and we give presentations for the automorphism groups of GL(2, Z), SL(2, L), PGL(2, Z) and PSL(2, Z), showing the close relationships between them.

A complete list is given of the conjugacy classes of PSL(2, Z)and PGL(2, Z), and it is seen that most are not invariant under the automorphisms of their groups. Congruence subgroups of PGL(2, Z) and PSL(2, Z) are studied, and we show that most are not characteristic in PGL(2, Z). In fact only a finite number are characteristic, demonstrating that their definition is essentially number-theoretic rather than group theoretic. 

By considering the theory of maps on surfaces we find a geometric interpretation for the outer automorphisms of PGL(2, Z). We show that the outer automorphism group of PGL(2, Z) acts on the class of trivalent maps to interchange faces and Petrie polygons.

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