G-Sets and Sequences Associated with nth Order Linear Recurrences Modulo Primes}

We consider nth order linear recurrence relations of the form $S_k=a_{k-1}S_{k-1}+a_{k-2}S_{k-2}+\cdots+a_{k-n}S_{k-n}$ over the finite field $Z_p$, where $p$ is a prime not equal to 2. The results regarding the distribution of elements in the sequence $\{S_0,S_1, \dots \}$ are well known for second order linear recurrence relations, however, we expand some results using matrix groups, linear algebra and $G$-sets in the finite vector space $\left(Z_p\right)^k$. It is our hope that this alternate approach may provide a set of material or examples that could be utilized in undergraduate mathematics courses.