# Isometry group of Minkowski Space

from the book "Quantum Field theory and The Standard Model - Schwartz M.D":

"The group of translations and Lorentz transformations is called the Poincare group $ISO(1,3)$ (the isometry group of Minkowski space)."

I understand that Poincare group consists of translations and Lorentz transformation, but what is $ISO(1,3)$? What does isometry group of Minkowski space means?

There are two questions. First $ISO(1,3)$ as a notation for the Poincare group is confusing/misleading and should normally be avoided. The "I" is meant for "inhomogenous" because in the time of Wigner and Bargmann, the Poincare group was called the "inhomogenous Lorentz group"
As you should know, there are three Lorentz groups, $O(1,3)$ - the full Lorentz group, this is traditionally denoted by $\mathcal L$, then $SO(1,3)$ - the Lorentz group of transformations with det =1, this is traditionally denoted by $\mathcal L_+$, then the so-called restricted Lorentz group, $SO^\uparrow (1,3)$, also denoted by $\mathcal L_{+}^{\uparrow}$. To each of the three groups, one defines their action on the $\mathbb R^4$ manifold of the abelian translations group and from here forms 3 semidirect products.
In physics, the isometry group of Minkowski spacetime is obviously the full Poincare group $\mathcal P = \mathbb R^4 \rtimes \mathcal L$, but this is too big for our purposes (Standard Model) because this contains the discrete PT transformations which can be individually broken. Therefore, the isometry group whose representations define the fundamental particles is only the subgroup $\mathcal P_{+}^{\uparrow} = \mathbb R \rtimes \mathcal L_{+}^{\uparrow}$.
$ISO(3,1)$ is not named after the word isometry. Rather it stands for inhomogeneous $O(3,1)$ or inhomogeneous Lorentz group, i.e. the group of inhomogeneous Lorentz transformations, aka. the Poincare group. It is the group of spacetime transformations that respect the Minkowski metric.