Timeline for Finite groups of order $n$ having exactly $n$ subgroups
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2015 at 5:42 | vote | accept | Joachim Grieg | ||
| Dec 4, 2015 at 13:28 | answer | added | Stefan Kohl♦ | timeline score: 7 | |
| Dec 4, 2015 at 8:08 | comment | added | M. Farrokhi D. G. | $C_4\times C_2$ is such a group. It seems $1$, $C_2$ and $C_4\times C_2$ are the only abelian groups with the mentioned property. | |
| S Dec 4, 2015 at 7:31 | history | suggested | CommunityBot | CC BY-SA 3.0 |
A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with this property?
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| Dec 4, 2015 at 6:58 | review | Suggested edits | |||
| S Dec 4, 2015 at 7:31 | |||||
| Dec 4, 2015 at 5:48 | comment | added | Gerry Myerson | Numbers $n$ such that $\sigma(n) + \tau(n) = 2n$ are tabulated at oeis.org/A083874 up to the 24th such $n$, which is 91707741184. | |
| Dec 4, 2015 at 4:46 | comment | added | M. Farrokhi D. G. | @GerryMyerson, Yes, of course! | |
| Dec 4, 2015 at 4:35 | comment | added | Gerry Myerson | @M.FarrokhiD.G., 130 is not of the form $2^kp$, $p$ prime. | |
| Dec 4, 2015 at 4:34 | comment | added | M. Farrokhi D. G. | A better question, in my opinion, is that whether there are infinitely many of such groups? | |
| Dec 4, 2015 at 4:33 | comment | added | M. Farrokhi D. G. | A dihedral group $D_{2n}$ has $\tau(n)+\sigma(n)$ subgroups. Finding those numbers $n$ satisfying the equation $\tau(n)+\sigma(n)=2n$ is itself a difficult problem. The values of $n\leq 10^7$ for which the above identity holds are $1$, $3$, $14$, $52$, $130$, $184$, $656$, $8648$, $12008$, $34688$ and $2118656$ among which $14$, $52$, $130$, $184$, $656$, $34688$ and $2118656$ are of the form $2^kp$ for some odd prime $p$. | |
| Dec 4, 2015 at 4:25 | comment | added | Joachim Grieg | Thanks! By looking to oeis.org/A018216, I found yet another example: a group of order 28. | |
| Dec 4, 2015 at 3:58 | comment | added | Yoav Kallus | Reminds me of this recent question: mathoverflow.net/questions/224636/…. Remeber: two is a coincidence; it takes three to make a trend. | |
| Dec 4, 2015 at 3:54 | review | Low quality posts | |||
| Dec 4, 2015 at 3:59 | |||||
| Dec 4, 2015 at 3:46 | comment | added | Joachim Grieg | Yes! I found only few finite groups with this property: the trivial group, $\mathbb{Z}_2$, $S_3$, ... and so on. | |
| Dec 4, 2015 at 3:39 | comment | added | Qiaochu Yuan | Any particular reason you want these two numbers to be the same? | |
| Dec 4, 2015 at 3:39 | review | First posts | |||
| Dec 4, 2015 at 4:05 | |||||
| Dec 4, 2015 at 3:34 | history | asked | Joachim Grieg | CC BY-SA 3.0 |