# Thomas precession and Lorentz group

I've recently learned about the Thomas precession. To shallowly summarize the Thomas precession takes place when you have a boost of first system to second system and then from the second system you have another non-collinear boost to the first one to a third system. If you now look at the relation between the first system and third system, the relation you get is just a different boost but you also have a rotation part of the transformation. Mathematically we could say $$\Lambda_{1 \rightarrow 2} \Lambda_{1 \rightarrow 2} \neq \Lambda_{1 \rightarrow 3}$$ where we denoted $$\Lambda$$ as the boost transformation and the subscript of $$\Lambda$$ denotes from what to which system we do the transformation. It can be shown rather that $$\Lambda_{1 \rightarrow 2} \Lambda_{1 \rightarrow 2} = R \Lambda \tag{1}$$ where $$R$$ denotes a rotation. This is all well and fine the problem that I have is that I've always believed Lorentz transformations $$\Lambda$$ are part of a (Lorentz) group. But if the relation (1) hold, this would break the definition of a group , specifically two objects in the group acting on each other would produce a object, which is not part of the group (Closure).

Could anyone please clear this up for me? Are perhaps rotations also part of the Lorentz group?