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.