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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1911.03564 |
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| _version_ | 1866910887031341056 |
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| author | de Oliveira, Leandro Nery de Alcântara, Marcos Aurélio |
| author_facet | de Oliveira, Leandro Nery de Alcântara, Marcos Aurélio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_03564 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Fubini Theorem for Normal Lie Subgroups of Index $2n$ de Oliveira, Leandro Nery de Alcântara, Marcos Aurélio Representation Theory 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. |
| title | The Fubini Theorem for Normal Lie Subgroups of Index $2n$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/1911.03564 |