Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915124503117824 |
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| author | Burkhart, Michael C. |
| author_facet | Burkhart, Michael C. |
| contents | We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16616 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions Burkhart, Michael C. Group Theory 20E22, 20F16, 20E45 (Primary), 05E18 (Secondary) We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma. |
| title | Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions |
| topic | Group Theory 20E22, 20F16, 20E45 (Primary), 05E18 (Secondary) |
| url | https://arxiv.org/abs/2211.16616 |