Fibonacci Group

In mathematics, for a natural number $n \ge 2$, the ''n''th 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:
* $a_1 a_2 = a_3,$
* $a_2 a_3 = a_4,$
* $\dots$
* $a_ a_ = a_n,$
* $a_a_n = a_1,$
* $a_n a_1 = a_2$.

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 /math> is defined as the set of all finite formal $K$-linear combinations of elements of $G$ − that is, an element $a$ of K /math> is of the form $a = \sum_ \lambda_g g$, 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 /math>, denoted $, \operatorname a\,,$, 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 /math> 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 $\left(\sum\nolimits_g \lambda_g g\right)\!\!\left(\sum\nolimits_h \mu_h h\right) = \sum\nolimits_ \lambda_g\mu_h \, gh$, whose support is again finite, and which can be written in the form $\sum_ \nu_x x$ as $\sum_\Bigg(\sum_ \lambda_g\mu_h \!\Bigg) x$.

'' 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 /math> does not contain any non-trivial units – that is, if $ab = 1$ in K /math> then $a = kg$ for some $k \in K$ and $g \in G$. Giles Gardam disproved this conjecture in February 2021 by providing a counterexample. He took $K = \mathbb_2$, 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_2 (2, 6)/math> he discovered has $, \operatorname \alpha\,, = , \operatorname \alpha^, = 21$.

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''.

References

External links

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
Group theory
Ring theory