In mathematics, for a natural number n ≥ 2 {\displaystyle n\geq 2}, the nth Fibonacci group, denoted F ( 2 , n ) {\displaystyle F(2,n)} or sometimes F ( n ) {\displaystyle F(n)}, is defined by n generators a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\dots ,a_{n}} and n relations:

  • a 1 a 2 = a 3 , {\displaystyle a_{1}a_{2}=a_{3},}
  • a 2 a 3 = a 4 , {\displaystyle a_{2}a_{3}=a_{4},}
  • … {\displaystyle \dots }
  • a n − 2 a n − 1 = a n , {\displaystyle a_{n-2}a_{n-1}=a_{n},}
  • a n − 1 a n = a 1 , {\displaystyle a_{n-1}a_{n}=a_{1},}
  • a n a 1 = a 2 {\displaystyle a_{n}a_{1}=a_{2}}.

These groups were introduced by John Conway in 1965.

The group F ( 2 , n ) {\displaystyle F(2,n)} is of finite order for n = 2 , 3 , 4 , 5 , 7 {\displaystyle n=2,3,4,5,7} and infinite order for n = 6 {\displaystyle n=6} and n ≥ 8 {\displaystyle n\geq 8}. The infinitude of F ( 2 , 9 ) {\displaystyle F(2,9)} was proved by computer in 1990.

Kaplansky's unit conjecture

From a group G {\displaystyle G} and a field K {\displaystyle K} (or more generally a ring), the group ring K [ G ] {\displaystyle K[G]} is defined as the set of all finite formal K {\displaystyle K}-linear combinations of elements of G {\displaystyle G} − that is, an element a {\displaystyle a} of K [ G ] {\displaystyle K[G]} is of the form a = ∑ g ∈ G λ g g {\displaystyle a=\sum _{g\in G}\lambda _{g}g}, where λ g = 0 {\displaystyle \lambda _{g}=0} for all but finitely many g ∈ G {\displaystyle g\in G} so that the linear combination is finite. The (size of the) support of an element a = ∑ g λ g g {\displaystyle a=\sum \nolimits _{g}\lambda _{g}g} in K [ G ] {\displaystyle K[G]}, denoted | supp ⁡ a | {\displaystyle |\operatorname {supp} a\,|}, is the number of elements g ∈ G {\displaystyle g\in G} such that λ g ≠ 0 {\displaystyle \lambda _{g}\neq 0}, i.e. the number of terms in the linear combination. The ring structure of K [ G ] {\displaystyle K[G]} is the "obvious" one: the linear combinations are added "component-wise", i.e. as ∑ g λ g g + ∑ g μ g g = ∑ g ( λ g + μ g ) g {\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 ( ∑ g λ g g ) ( ∑ h μ h h ) = ∑ g , h λ g μ h g h {\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 ∑ x ∈ G ν x x {\displaystyle \sum _{x\in G}\nu _{x}x} as ∑ x ∈ G ( ∑ g , h ∈ G g h = x λ g μ h ) x {\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 K {\displaystyle K} and a torsion-free group G {\displaystyle G} (a group in which all non-identity elements have infinite order), the group ring K [ G ] {\displaystyle K[G]} does not contain any non-trivial units – that is, if a b = 1 {\displaystyle ab=1} in K [ G ] {\displaystyle K[G]} then a = k g {\displaystyle a=kg} for some k ∈ K {\displaystyle k\in K} and g ∈ G {\displaystyle g\in G}. Giles Gardam disproved this conjecture in February 2021 by providing a counterexample. He took K = F 2 {\displaystyle K=\mathbb {F} _{2}}, the finite field with two elements, and he took G {\displaystyle G} to be the 6th Fibonacci group F ( 2 , 6 ) {\displaystyle F(2,6)}. The non-trivial unit α ∈ F 2 [ F ( 2 , 6 ) ] {\displaystyle \alpha \in \mathbb {F} _{2}[F(2,6)]} he discovered has | supp ⁡ α | = | supp ⁡ α − 1 | = 21 {\displaystyle |\operatorname {supp} \alpha \,|=|\operatorname {supp} \alpha ^{-1}|=21}.

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

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