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.

In this paper, we introduce the concepts of 2-isometry, collinearity, 2-Lipschitzmapping in 2-fuzzy 2-normed linear spaces. Also, we give a new generalization of the Mazur- Ulamtheorem when X is a 2-fuzzy 2-normed linear space or Á(X) is a fuzzy 2-normed linear space, that is, the Mazur-Ulam theorem holds, when the 2-isometry mapped to a 2-fuzzy 2-normedlinear space is affine.

The aim of this work is to characterize all bounded linear operators $T:\lpi\rightarrow\lpi$ which preserve disjoint support property. We show that the constant coefficients of all isometries on $\lpi$ are in the class of such operators, where $2\neq p\in [1,\infty )$ and $I$ is a non-empty set. We extend preserving disjoint support property to linear operators on $\mathfrak{c}_{0}(I).$ At the end, we obtain some equivalent properties of isometries on Banach spaces.