Fibonacci group

In mathematics, for a natural number $n \ge 2$ , the nth Fibonacci group, denoted $F(2,n)$ or sometimes $F(n)$ , is defined by n generators $a_1, a_2, \dots, a_n$ and n relations:

$a_1 a_2 = a_3,$
$a_2 a_3 = a_4,$
$\dots$
$a_{n-2} a_{n-1} = a_n,$
$a_{n-1}a_n = a_1,$
$a_n a_1 = a_2$ .

These groups were introduced by John Conway in 1965.

The group $F(2,n)$ is of finite order for $n=2,3,4,5,7$ and infinite order for $n = 6$ and $n \ge 8$ .

The infinitude of $F(2,9)$ was proved by computer in 1990.

Kaplansky's unit conjecture

From a group $G$ and a field $K$ (or more generally a ring), the group ring $K[G]$ is defined as the set of all finite formal $K$ -linear combinations of elements of $G$ − that is, an element $a$ of $K[G]$ is of the form $a = \sum_{g \in G} \lambda_g g$ , where $\lambda_g = 0$ for all but finitely many $g \in G$ so that the linear combination is finite. The (size of the) support of an element $a = \sum\nolimits_g \lambda_g g$ in $K[G]$ , denoted $|\operatorname{supp} a\,|$ , is the number of elements $g \in G$ such that $\lambda_g \neq 0$ , i.e. the number of terms in the linear combination. The ring structure of $K[G]$ is the "obvious" one: the linear combinations are added "component-wise", i.e. as $\sum\nolimits_g \lambda_g g + \sum\nolimits_g \mu_g g = \sum\nolimits_g (\lambda_g \!+\! \mu_g) g$ , whose support is also finite, and multiplication is defined by $\left(\sum\nolimits_g \lambda_g g\right)\!\!\left(\sum\nolimits_h \mu_h h\right) = \sum\nolimits_{g,h} \lambda_g\mu_h \, gh$ , whose support is again finite, and which can be written in the form $\sum_{x \in G} \nu_x x$ as $\sum_{x \in G}\Bigg(\sum_{g,h \in G \atop gh = x} \lambda_g\mu_h \!\Bigg) x$ .

Kaplansky's unit conjecture states that given a field $K$ and a torsion-free group $G$ (a group in which all non-identity elements have infinite order), the group ring $K[G]$ does not contain any non-trivial units – that is, if $ab = 1$ in $K[G]$ then $a = kg$ for some $k \in K$ and $g \in G$ . Giles Gardam disproved this conjecture in February 2021 by providing a counterexample.{{cite journal |last1=Gardam |first1=Giles |title=A counterexample to the unit conjecture for group rings |journal=Annals of Mathematics |year=2021 |volume=194 |issue=3 |doi=10.4007/annals.2021.194.3.9 |arxiv=2102.11818 |s2cid=232013430 }}{{cite web |title=Interview with Giles Gardam |url=https://www.uni-muenster.de/MathematicsMuenster/news/artikel/2021-03-04.shtml |publisher=Mathematics Münster, University of Münster |access-date=10 March 2021}}{{cite web |last1=Klarreich |first1=Erica |title=Mathematician Disproves 80-Year-Old Algebra Conjecture |url=https://www.quantamagazine.org/mathematician-disproves-group-algebra-unit-conjecture-20210412/ |publisher=Quanta Magazine |access-date=13 April 2021}} He took $K = \mathbb{F}_2$ , the finite field with two elements, and he took $G$ to be the 6th Fibonacci group $F(2,6)$ . The non-trivial unit $\alpha \in \mathbb{F}_2[F(2, 6)]$ he discovered has $|\operatorname{supp} \alpha\,| = |\operatorname{supp} \alpha^{-1}| = 21$ .

The 6th Fibonacci group $F(2,6)$ has also been variously referred to as the Hantzsche-Wendt group, the Passman group, and the Promislow group.{{cite web |last1=Gardam |first1=Giles |title=Kaplansky's conjectures |website=YouTube |url=https://www.youtube.com/watch?v=DJEabykk8j8}}

References

External links

[http://www.expmath.org/restricted/4/4.2/holt.ps An alternative proof that the Fibonacci group F(2,9) is infinite] by Derek K. Holt (PostScript file).
{{Cite web|title=Fibonacci group|url=https://encyclopediaofmath.org/wiki/Fibonacci_group|website=Encyclopedia of Mathematics}}

Category:Group theory

Category:Ring theory