New lower bounds for the number of conjugacy classes in finite nilpotent groups
P. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2022-06-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | https://ijgt.ui.ac.ir/article_25810_6cf96773007f3bab56c3708c2139b4e7.pdf |
| _version_ | 1818487556665245696 |
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| author | Edward A. Bertram |
| author_facet | Edward A. Bertram |
| author_sort | Edward A. Bertram |
| collection | DOAJ |
| description | P. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/3$, which is best possible across all nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| \ne 8,16$. These results are then used to prove that $k(G) > \log_3(|G|)$ when $G/N$ is nilpotent, under natural conditions on $N \trianglelefteq G$. Also, when $G'$ is nilpotent of class $c$, we prove that $k(G) \ge (\log |G|)^t$ when $|G|$ is large enough, depending only on $(c,t)$. |
| first_indexed | 2024-12-10T16:39:29Z |
| format | Article |
| id | doaj.art-d937c2cfb81b401e9cf2dbb2149743a8 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-10T16:39:29Z |
| publishDate | 2022-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-d937c2cfb81b401e9cf2dbb2149743a82022-12-22T01:41:16ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692022-06-0111210911910.22108/ijgt.2021.128396.168725810New lower bounds for the number of conjugacy classes in finite nilpotent groupsEdward A. Bertram0Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USAP. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/3$, which is best possible across all nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| \ne 8,16$. These results are then used to prove that $k(G) > \log_3(|G|)$ when $G/N$ is nilpotent, under natural conditions on $N \trianglelefteq G$. Also, when $G'$ is nilpotent of class $c$, we prove that $k(G) \ge (\log |G|)^t$ when $|G|$ is large enough, depending only on $(c,t)$.https://ijgt.ui.ac.ir/article_25810_6cf96773007f3bab56c3708c2139b4e7.pdfnilpotentconjugacyderived series |
| spellingShingle | Edward A. Bertram New lower bounds for the number of conjugacy classes in finite nilpotent groups International Journal of Group Theory nilpotent conjugacy derived series |
| title | New lower bounds for the number of conjugacy classes in finite nilpotent groups |
| title_full | New lower bounds for the number of conjugacy classes in finite nilpotent groups |
| title_fullStr | New lower bounds for the number of conjugacy classes in finite nilpotent groups |
| title_full_unstemmed | New lower bounds for the number of conjugacy classes in finite nilpotent groups |
| title_short | New lower bounds for the number of conjugacy classes in finite nilpotent groups |
| title_sort | new lower bounds for the number of conjugacy classes in finite nilpotent groups |
| topic | nilpotent conjugacy derived series |
| url | https://ijgt.ui.ac.ir/article_25810_6cf96773007f3bab56c3708c2139b4e7.pdf |
| work_keys_str_mv | AT edwardabertram newlowerboundsforthenumberofconjugacyclassesinfinitenilpotentgroups |