Fibonacci group

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

Kaplansky's unit conjecture states that given a field ${\displaystyle K}$ and a torsion-free group ${\displaystyle G}$ (a group in which all non-identity elements have infinite order), the group ring ${\displaystyle K[G]}$ does not contain any non-trivial units – that is, if ${\displaystyle ab=1}$ in ${\displaystyle K[G]}$ then ${\displaystyle a=kg}$ for some ${\displaystyle k\in K}$ and ${\displaystyle g\in G}$. Giles Gardam disproved this conjecture in February 2021 by providing a counterexample.[1][2][3] He took ${\displaystyle K=\mathbb {F} _{2}}$, the finite field with two elements, and he took ${\displaystyle G}$ to be the 6th Fibonacci group ${\displaystyle F(2,6)}$. The non-trivial unit ${\displaystyle \alpha \in \mathbb {F} _{2}[F(2,6)]}$ he discovered has ${\displaystyle |\operatorname {supp} \alpha \,|=|\operatorname {supp} \alpha ^{-1}|=21}$.[1]

The 6th Fibonacci group ${\displaystyle F(2,6)}$ has also been variously referred to as the Hantzsche-Wendt group, the Passman group, and the Promislow group.[1][4]

• An alternative proof that the Fibonacci group F(2,9) is infinite by Derek K. Holt (PostScript file).
• "Fibonacci group". Encyclopedia of Mathematics.