Fibonacci group

In mathematics, for a natural number

, the nth Fibonacci group, denoted

, is defined by n generators

and n relations: These groups were introduced by John Conway in 1965.

The group

is of finite order for

and infinite order for

was proved by computer in 1990.

(or more generally a ring), the group ring

is defined as the set of all finite formal

-linear combinations of elements of

so that the linear combination is finite.

The (size of the) support of an element

, is the number of elements

, i.e. the number of terms in the linear combination.

The ring structure of

is the "obvious" one: the linear combinations are added "component-wise", i.e. as

, whose support is also finite, and multiplication is defined by

, whose support is again finite, and which can be written in the form

Kaplansky's unit conjecture states that given a field

and a torsion-free group

(a group in which all non-identity elements have infinite order), the group ring

does not contain any non-trivial units – that is, if

Giles Gardam disproved this conjecture in February 2021 by providing a counterexample.

, the finite field with two elements, and he took

to be the 6th Fibonacci group

The non-trivial unit

supp ⁡ α

[1] The 6th Fibonacci group

has also been variously referred to as the Hantzsche-Wendt group, the Passman group, and the Promislow group.