Commuting probability for conjugate subgroups of a finite group
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915288640913408 |
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| author | Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel |
| author_facet | Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel |
| contents | Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question.
Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in G. Is the order of P modulo O_p(G) bounded in terms of e only?
With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established.
We also show that if \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Commuting probability for conjugate subgroups of a finite group Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel Group Theory 20D20, 20E45, 20P05 Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in G. Is the order of P modulo O_p(G) bounded in terms of e only? With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G. |
| title | Commuting probability for conjugate subgroups of a finite group |
| topic | Group Theory 20D20, 20E45, 20P05 |
| url | https://arxiv.org/abs/2505.10521 |