The Fubini Theorem for Normal Lie Subgroups of Index $2n$
Let $Γ_+$ be a normal subgroup of index $2n$ of a group $Γ$ and $γ_i \in Γ\setminus Γ_+$ be involutions. We first prove that if $Γ= Γ_+ \rtimes (\mathbb{Z}_2(γ_1) \times \cdots \times \mathbb{Z}_2(γ_n))$ then $Γ= (Γ_+ \rtimes \mathbb{Z}_2(γ_1) \rtimes \cdots \rtimes \mathbb{Z}_2(γ_{i-1})) \rtimes (\mathbb{Z}_2(γ_{i}) \times \cdots \times \mathbb{Z}_2(γ_n))$, where $i=2,\cdots,n$. Second, we use this result to prove the well-known Fubini theorem for a subgroup of index $2n$ of a compact Lie group. Finally, we present an application to invariant theory.
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