Timeline for Measures of non-abelian-ness
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2019 at 6:59 | comment | added | MathematicsStudent1122 | @ArturoMagidin My apologies. Overlooking what was mentioned in the answer was quite foolish of me. Thanks for the response. | |
| Jan 13, 2019 at 6:30 | comment | added | Arturo Magidin | @MathematicsStudent1122: Look: if I tell you I have a group with trivial center, that doesn’t really tell you how likely two elements are to commute. If I tell you I have a group with $G=G’$, that also does not tell you much about how likely two elements are to commute; worse if I tell you “I have a group with commutator subgroup of order $1000$”. That’s why I said that neither the center nor the commutator subgroups are good quantitative measures of how “abelian” a group is. Six years ago... | |
| Jan 13, 2019 at 6:26 | comment | added | Arturo Magidin | @MathematicsStudent1122: $P(G)$ cannot exceed $\frac{5}{8}$ in a noncommutative group, as it says in the answer. So the answer is obviously “no”, because if you pick $\epsilon$ too small, then $Z(G)=G$. | |
| Jan 13, 2019 at 6:19 | comment | added | MathematicsStudent1122 | @ArturoMagidin With regard to the very last point you just made, is it true that for any small $\epsilon, \delta>0$ we can find a group $G$ with $P(G) > 1 - \epsilon$ but $\frac{|Z(G)|}{|G|} < \delta$? In other words, "small centre" but "lots and lots of things commute". Is there a concrete example of this? | |
| Jan 13, 2019 at 6:12 | comment | added | Arturo Magidin | @MathematicsStudent1122: The size of the groups doesn’t give you a good gauge of how likely two elements are to commute. You could have a very “nonabelian” group with very few commutators, quantitatively speaking, or a group with very small center but where lots and lots of things commute with one another. | |
| Jan 13, 2019 at 5:58 | comment | added | MathematicsStudent1122 | Could you elaborate on the first sentence of this answer? Why are $Z(G)$ and $[G,G]$ not good quantitative measures? | |
| Sep 10, 2018 at 15:11 | comment | added | LSpice | Links: MacHale (MSN); Rusin (MSN); Guralnick and Robinson and erratum (MSN). | |
| Mar 25, 2013 at 18:26 | history | edited | Arturo Magidin | CC BY-SA 3.0 |
fix accent in Erdős, add MacHale, Rusin, Guralnick-Robinson reference
|
| Mar 25, 2013 at 10:48 | comment | added | Joseph O'Rourke | @Arturo: TheTurán-Erdős model is perfect--Thanks so much! Fascinating that certain values of $P(G)$ cannot occur. | |
| Mar 25, 2013 at 9:56 | vote | accept | Joseph O'Rourke | ||
| Mar 25, 2013 at 0:52 | history | edited | Arturo Magidin | CC BY-SA 3.0 |
added 182 characters in body; added 120 characters in body
|
| Mar 25, 2013 at 0:46 | history | answered | Arturo Magidin | CC BY-SA 3.0 |