The Generalized Order- Lucas Sequences in Finite Groups

We study the generalized order- Lucas sequences modulo . Also, we define the th generalized order- Lucas orbit ,{1,2,…,？1}() with respect to the generating set and the constants 1,2, and ？1 for a finite group =？？. Then, we obtain the lengths of the periods of the th generalized order- Lucas orbits of the binary polyhedral groups ？,2,2？,？2,,2？,？2,2,？ and the polyhedral groups (,2,2),(2,,2),(2,2,) for 1≤≤.