Timeline for A generalization of $p$-groups [closed]
Current License: CC BY-SA 4.0
9 events
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| Jul 31, 2019 at 13:47 | history | closed |
YCor Victor Protsak Derek Holt Pace Nielsen LSpice |
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| Jul 19, 2019 at 9:14 | comment | added | Victor Protsak | A better generalization is given by finite nilpotent groups. Every finite $p$-group is nilpotent and by a well- known theorem, each finite nilpotent group is isomorphic to the direct product of its Sylow subgroups. | |
| Jul 19, 2019 at 8:57 | comment | added | YCor | An example of a reference is Burnside's $p^aq^b$ Theorem (Wikipedia link). | |
| Jul 19, 2019 at 8:25 | review | Close votes | |||
| Jul 31, 2019 at 13:47 | |||||
| Jul 19, 2019 at 8:15 | comment | added | zibadawa timmy | Seems unlikely to admit a fairly general approach, because you'd be studying all groups of order $m$ of necessity (as your class would include the groups $G^k$ for any $|G|=m$), and that can get very ad hoc and complex if $m$ isn't very nice. And just about every property that makes $p$-groups study-able in (more-or-less) their entirety are all dependent on the primality. Are there particular things you're hoping to know about such groups? | |
| Jul 19, 2019 at 8:06 | history | edited | YCor | CC BY-SA 4.0 |
added 10 characters in body; edited title
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| Jul 19, 2019 at 8:05 | comment | added | YCor | I guess the right generalization would be to study classes of finite groups in the generated by a group variety under taking extensions. A closer case would be to consider all groups with prime divisors in some given finite set, e.g., all groups of order dividing $6^n$ for some $n$. Indeed the class you're suggesting is not stable under taking subgroups/quotients when $m$ is not a prime power. | |
| Jul 19, 2019 at 6:52 | history | edited | user64494 | CC BY-SA 4.0 |
Typos are corrected.
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| Jul 19, 2019 at 5:43 | history | asked | Mohammad Radi | CC BY-SA 4.0 |