On \((2,n)\)-groups related to Fibonacci groups
The authors study the groups \(G(n;h,i,j,k)\) with presentations of the form \[ <a,b: a^ 2=b^ n=ab^ hab^ iab^ jab^ k=1> \] where \(h+i+j+k=0\) and \(h,i,j,k\) are in \(\{1,-1,2,-2\}.\) To determine which of these groups are finite it may be assumed that \(h=1\) and if \[ G(n;i,j,k)=<a,b: a^ 2=b^ n=abab^ iab^ jab^ k=1>, \] then the problem is reduced to the study of five cases. It is shown that: (1) \(G(n;1,-1,-1)\) is metabelian of order \(2n^ 2\). (2) \(G(n;2,-1,-2) \cong G(n;1,-1,-1) (n odd);\) \(G(n;2,-1,-2)\) is infinite (n even); (3) \(G(n;-1,1,-1) \cong D_ 8\) \((n=2)\); \(G(n;-1,1,-1) \cong A_ 4\times C_ 2\) \((n=3)\); \(G(n;-1,1,-1)\) is infinite \((n>3)\); (4) \(G(n;-2,2,-1)\) is metabelian of order \(2n(2^ n-(-1)^ n)/3\). (5) \(G(n;2,-2,-1) \cong E_ 4\) \((n=2)\); \(G(n;2,-2,-1)\) is metabelian of order 40 \((n=4)\); \(G(n;2,-2,-1)\) is infinite (n even, \(n>4)\); \(G(n;2,-2,- 1)\) is of order \(2ng_ n\) (n odd) where \((g_ n)\) denotes the Lucas sequence of numbers.
scientific article; zbMATH DE number 4029820