2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229265Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is linear. This result gives a large linear subgroup of Aut(F_3). A second application is that the mapping class group for genus one surfaces with two punctures is linear.Group TheoryGeometric Topology20F28text