Fibonacci group explained

n \ge 2 , the nth Fibonacci group, denoted F( 2 ,n) or sometimes F(n) , is defined by n generators a _{1,} a _{2,} ... , a _{n} and n relations:a _{1} a _{2} = a _{3,} a _{2} a _{3} = a _{4,} ... 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

See also: Kaplansky's conjectures.

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} λ _{g} g , where λ _{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} λ _{g} g in K[G] , denoted |\operatorname{supp} a| , is the number of elements g \in G such that λ _{g} ≠ 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} λ _{g} g + \sum\nolimits _{g} \mu _{g} g = \sum\nolimits _{g} ( λ _{g} + \mu _{g)} g , whose support is also finite, and multiplication is defined by \left(\sum\nolimits _{g} λ _{g} g\right)\left(\sum\nolimits _{h} \mu _{h} h\right) = \sum\nolimits _{g,h} λ _{g\mu} _{h} gh , whose support is again finite, and which can be written in the form \sum _{x} \nu _{x} x as \sum _{x} ( \sum _{g,h} λ _{g\mu} _{h} ) 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.^{[1] [2] [3]} 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 |\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.^[4]

External links

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

Notes and References

1. Gardam . Giles . A counterexample to the unit conjecture for group rings . Annals of Mathematics . 2021 . 194 . 3 . 10.4007/annals.2021.194.3.9 . 2102.11818 . 232013430 .
2. Web site: Interview with Giles Gardam . . 10 March 2021.
3. Web site: Klarreich . Erica . Mathematician Disproves 80-Year-Old Algebra Conjecture . . 13 April 2021.
4. Web site: Gardam . Giles . Kaplansky's conjectures . .