In mathematics, for a natural number formula_1, the "n"th Fibonacci group, denoted formula_2 or sometimes formula_3, is defined by "n" generators formula_4 and "n" relations:
These groups were introduced by John Conway in 1965.
The group formula_2 is of finite order for formula_12 and infinite order for formula_13 and formula_14. 
The infinitude of formula_15 was proved by computer in 1990.
Kaplansky's unit conjecture.
From a group formula_16 and a field formula_17 (or more generally a ring), the group ring formula_18 is defined as the set of all finite formal formula_17-linear combinations of elements of formula_16 − that is, an element formula_21 of formula_18 is of the form formula_23, where formula_24 for all but finitely many formula_25 so that the linear combination is finite. The (size of the) "support" of an element formula_26 in formula_18, denoted formula_28, is the number of elements formula_25 such that formula_30, i.e. the number of terms in the linear combination. The ring structure of formula_18 is the "obvious" one: the linear combinations are added "component-wise", i.e. as formula_32, whose support is also finite, and multiplication is defined by formula_33, whose support is again finite, and which can be written in the form formula_34 as formula_35.
"Kaplansky's unit conjecture" states that given a field formula_17 and a torsion-free group formula_16 (a group in which all non-identity elements have infinite order), the group ring formula_18 does not contain any non-trivial units – that is, if formula_39 in formula_18 then formula_41 for some formula_42 and formula_25. Giles Gardam disproved this conjecture in February 2021 by providing a counterexample. He took formula_44, the finite field with two elements, and he took formula_16 to be the 6th Fibonacci group formula_46. The non-trivial unit formula_47 he discovered has formula_48.
The 6th Fibonacci group formula_46 has also been variously referred to as the "Hantzsche-Wendt group", the "Passman group", and the "Promislow group".
References.
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