In mathematics, for a natural number , the nth Fibonacci group, denoted or sometimes , 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 and . The infinitude of was proved by computer in 1990.

Kaplansky's unit conjecture

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From a group   and a field   (or more generally a ring), the group ring   is defined as the set of all finite formal  -linear combinations of elements of   − that is, an element   of   is of the form  , where   for all but finitely many   so that the linear combination is finite. The (size of the) support of an element   in  , denoted  , is the number of elements   such that  , 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   as  .

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   in   then   for some   and  . Giles Gardam disproved this conjecture in February 2021 by providing a counterexample.[1][2][3] He took  , the finite field with two elements, and he took   to be the 6th Fibonacci group  . The non-trivial unit   he discovered has  .[1]

The 6th Fibonacci group   has also been variously referred to as the Hantzsche-Wendt group, the Passman group, and the Promislow group.[1][4]

References

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  1. ^ a b c Gardam, Giles (2021). "A counterexample to the unit conjecture for group rings". Annals of Mathematics. 194 (3). arXiv:2102.11818. doi:10.4007/annals.2021.194.3.9. S2CID 232013430.
  2. ^ "Interview with Giles Gardam". Mathematics Münster, University of Münster. Retrieved 10 March 2021.
  3. ^ Klarreich, Erica. "Mathematician Disproves 80-Year-Old Algebra Conjecture". Quanta Magazine. Retrieved 13 April 2021.
  4. ^ Gardam, Giles. "Kaplansky's conjectures". YouTube.
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