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. 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.