Fibonacci group

Template:Use dmy dates 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 $a=\sum _g{\lambda }_{g}g$ in $K[G]$, denoted $|\mathrm{supp}a|$, is the number of elements $g\in G$ such that ${\lambda }_g\ne 0$, 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 $\sum _g{\lambda }_{g}g+\sum _g{\mu }_{g}g=\sum _g({\lambda }_g+{\mu }_g)g$, whose support is also finite, and multiplication is defined by $\left(\sum _g{\lambda }_{g}g\right)\left(\sum _h{\mu }_{h}h\right)=\sum _{g,h}{\lambda }_g{\mu }_{h}gh$, whose support is again finite, and which can be written in the form $\sum _{x\in G}{\nu }_{x}x$ as $\sum _{x\in G}\left(\sum _{\genfrac{}{}{0ex}{}{g,h\in G}{gh=x}}{\lambda }_g{\mu }_h\right)x$.

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 $K={𝔽}_2$, the finite field with two elements, and he took $G$ to be the 6th Fibonacci group $F(2,6)$. The non-trivial unit $\alpha \in {𝔽}_2[F(2,6)]$ he discovered has $|\mathrm{supp}\alpha |=|\mathrm{supp}{\alpha }^{-1}|=21$.

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.

External links

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