Hamid Reza Salimi Moghaddam - Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, ۸۱۷۴۶-۷۳۴۴۱-Iran.
Masumeh Nejadahmad - Department of Mathematics, Isfahan University of Technology, Iran.

Let F be an (α,β)-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group G. We consider the automorphism and isometry groups of the Finsler manifold (G,F) and their intersection. We prove that for an arbitrary left invariant vector field X and any compact subgroup K of automorphisms which X is invariant under them, there exists an (α,β)-metric such that K is a subgroup of its isometry group.

Automorphism group, Isometry group, Lie group, Left Invariant (\alpha, beta)-metric