Timeline for Minimal generation for finite abelian groups
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2011 at 11:18 | vote | accept | Calc | ||
| Dec 5, 2011 at 10:52 | vote | accept | Calc | ||
| Dec 5, 2011 at 10:52 | |||||
| Dec 4, 2011 at 21:41 | comment | added | Arturo Magidin | As noted in math.SE, $k\leq c$ holds: pick a prime $p$ that divides $d_1$. Then $G/pG$ is a vector space over $\mathbf{F}_p$ of dimension $k$, hence $G$ needs at least $k$ generators. | |
| Dec 4, 2011 at 21:38 | comment | added | Arturo Magidin | Also posted in math.SE: math.stackexchange.com/questions/88106/… | |
| Dec 4, 2011 at 13:13 | comment | added | Geoff Robinson | @Boris: Not sure that I agree with the above description of what $n$ is. The number of summands when $G$ is expressed as a direct sum of cyclic groups is not uniquely determined.For example, in general the number $k$ and $n$ above are usually different. | |
| Dec 4, 2011 at 13:08 | answer | added | Geoff Robinson | timeline score: 6 | |
| Dec 4, 2011 at 12:18 | comment | added | Boris Novikov | I don't know. :-( | |
| Dec 4, 2011 at 11:59 | comment | added | Calc | Ok. Sorry. How can one erase a question? ;) | |
| Dec 4, 2011 at 11:46 | comment | added | Boris Novikov | OK, so $n$ is the number of summands in the direct sum of cyclic groups, isn't it? Then the inequality $c\le n$ is evident. | |
| Dec 4, 2011 at 11:31 | comment | added | Calc | It is part of the fundamental theorem for finitely generated abelian groups that the $d_i, p_i, \alpha_i$ are uniquely determined by $G$ itself (up to reordering in the second case). If instead you are asking about a name, I heard of people calling $k$ and $n$ respectively the minimal and maximal rank. | |
| Dec 4, 2011 at 11:15 | comment | added | Boris Novikov | How do you define $n$? | |
| Dec 4, 2011 at 10:17 | history | edited | Calc | CC BY-SA 3.0 |
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| Dec 4, 2011 at 8:58 | history | edited | Calc | CC BY-SA 3.0 |
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| Dec 4, 2011 at 8:35 | history | asked | Calc | CC BY-SA 3.0 |