The generalized order (k,t)-Mersenne sequences in groups

Abstract

The purpose of this paper is to determine the algebraic properties of finite groups via a Mersenne-like sequence. Firstly, we introduce the generalized order $(k,t)$-Mersenne number sequences and study the periods of these sequences modulo $m$. Then, we get some interesting structural results. Furthermore, we expand the generalized order $(k,t)$-Mersenne number sequences to groups and we give the definition of the generalized order $(k,t)$-Mersenne sequences, $MQ_k^t(G,X)$, in the $j$-generator groups and also, investigate these sequences in the non-Abelian finite groups in detail. At last, we obtain the periods of the generalized order $(k,t)$-Mersenne sequences in some special groups as applications of the results produced.