Lorentz group and the restricted Lorentz group

[TrickyDicky]

It is a well known fact that the Lorentz group of transfornations are linear.
Now reading the wiki entry on the LG it spends a good deal explaining its identity component subgroup, the restricted LG group, and it turns out it is isomorphic to the linear fractional transformation group, which are non-linear transformations, now my doubt
(it might be silly) is how can a subgroup of linear transformations be nonlinear?

 

[Bill_K]

A group is defined abstractly by its group product. Or in the case of a continuous group, by its commutators. There's nothing to say whether the group is linear or not - linearity is a property of a particular representation. And as this example illustrates, the same group can have representations which are linear or nonlinear.

 

[TrickyDicky]

A group is defined abstractly by its group product. Or in the case of a continuous group, by its commutators. There's nothing to say whether the group is linear or not - linearity is a property of a particular representation. And as this example illustrates, the same group can have representations which are linear or nonlinear.

Thanks Bill, I was on my way to realizing just that, the proper orthochronous Lorentz transformations involve the matrix product of boosts and rotations both of which are linear but the product needs not be.