Groups with conjugacy classes of coprime sizes
Fuente:
arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915446878371840 |
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| author | Camina, Rachel D. Maróti, Attila Pacifici, Emanuele Parker, Chris Rekvényi, Kamilla Saunders, Jack Sotomayor, Víctor Tracey, Gareth van Beek, Martin |
| author_facet | Camina, Rachel D. Maróti, Attila Pacifici, Emanuele Parker, Chris Rekvényi, Kamilla Saunders, Jack Sotomayor, Víctor Tracey, Gareth van Beek, Martin |
| contents | Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Groups with conjugacy classes of coprime sizes Camina, Rachel D. Maróti, Attila Pacifici, Emanuele Parker, Chris Rekvényi, Kamilla Saunders, Jack Sotomayor, Víctor Tracey, Gareth van Beek, Martin Group Theory Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$. |
| title | Groups with conjugacy classes of coprime sizes |
| topic | Group Theory |
| url | https://arxiv.org/abs/2508.03851 |