Timeline for p-group with large center
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 5, 2016 at 12:25 | answer | added | yakov | timeline score: -4 | |
| Jul 5, 2016 at 12:19 | comment | added | yakov | In the my post above such $p$-groups are characterized so other atgument are superfluous. Indeed, let $|G:{\rm Z}(G)|\p^2$ and $S\le G$ minimal nonabelian. Set $H=S\text{Z}(G)$. As $|H:\text{Z}(G)|\ge p^2=|G:\text{Z}(G)|$, we get $H=G$. Conversely, if $G=S\text{Z}(G)$, where $S$ is minimal nonabelian, then $\text{Z}(S)\le\text{Z}(G)$ and we conclude that $|G:\text{Z}(G)|=|S:\text{Z}(S)|=p^2$. | |
| Jul 5, 2016 at 12:12 | comment | added | yakov | In my post above such groups are characterized. So all other argumant are auperfluous. Indeed, let $|G:\text{Z}(G)|=p^2$ and let $S\le G$ be minimal nonabelian. Set $H=S\text{Z}(G)$. As $H$ is nonabelian, $|H:\text{Z}(G)|\ge | |
| Jun 25, 2016 at 16:11 | comment | added | yakov | A $p$-group $G$ satisfies the condition iff $G=S{\rm Z}(G)$, where $S\le G$ is minimal nonabelian. | |
| Feb 25, 2013 at 4:39 | answer | added | Arturo Magidin | timeline score: 5 | |
| Feb 21, 2013 at 20:30 | comment | added | Hamid Shahverdi | Yes, You are right. | |
| Feb 21, 2013 at 19:59 | comment | added | Arturo Magidin | @Hamid: No, there's many such groups, obtained by varying $A$. E.g., with $n=5$, you can have $A=C_{p}^3$, yielding a group of exponent $p$, or $A=C_{p^3}$, yielding a group with an element of order $p^3$. | |
| Feb 21, 2013 at 19:39 | answer | added | Ralph | timeline score: 13 | |
| Feb 21, 2013 at 19:37 | comment | added | Hamid Shahverdi | Steve, For any integer $n\geq 4$ there exist only one group sa you make. | |
| Feb 21, 2013 at 19:16 | comment | added | Steve D | If $H$ is the p-group of exponent $p$ and order $p^3$, and $A$ is any abelian group, then any central product of $A$ and $H$ has center $A$ of index $p^2$. | |
| Feb 21, 2013 at 19:14 | history | edited | Alexander Chervov |
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| Feb 21, 2013 at 17:22 | history | asked | Hamid Shahverdi | CC BY-SA 3.0 |