The Fubini Theorem for Normal Lie Subgroups of Index $2n$

Abstract

Let ${\Gamma }_{+}$ be a normal subgroup of index $2n$ of a group $\Gamma$ and ${\gamma }_i\in \Gamma \setminus {\Gamma }_{+}$ be involutions. We first prove that if $\Gamma ={\Gamma }_{+}⋊({\mathbb{Z}}_2({\gamma }_1)\times \cdots \times {\mathbb{Z}}_2({\gamma }_n))$ then $\Gamma =({\Gamma }_{+}⋊{\mathbb{Z}}_2({\gamma }_1)⋊\cdots ⋊{\mathbb{Z}}_2({\gamma }_{i-1}))⋊({\mathbb{Z}}_2({\gamma }_i)\times \cdots \times {\mathbb{Z}}_2({\gamma }_n))$, 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.

arXiv comment12 pages

References in corpus (1)

• Equivariant mappings and invariant sets on Minkowski space