On automorphisms of a group of prime exponent (Q2711633)

\textit{D. D. Wall} [Am. Math. Mon. 67, 525-532 (1960; Zbl 0101.03201)] introduced the idea of Fibonacci sequences for groups. Let \(a\), \(b\) be two elements of a group, \(G\), and consider the sequence given by \(x_0=a\), \(x_1=b\), \(x_{n+1}=x_{n-1}x_n\). Wall examined the case when \(G\) was cyclic. \textit{H. Aydin} and \textit{G. C. Smith} [J. Lond. Math. Soc., II. Ser. 49, No. 1, 83-92 (1994; Zbl 0807.20029)] extended these results concerning the length of the fundamental period of the Fibonacci sequence to finite \(p\)-groups. It is clear that these results are intimately related to properties of 2-generator results. In the paper by Aydin and Smith they discuss the length of the Fibonacci sequence for \(R(2,5)\) where \(R(2,5)\) is the largest finite 2-generator group of exponent 5.NEWLINENEWLINENEWLINESuppose that \(G\) is a finite 2-generator group of prime exponent \(p\) and \(\phi\) is an automorphism of \(G\). The author shows that if the automorphism \(\phi\) induced on \(G/G'\) has minimal polynomial \((x-\lambda)^2\), where \(\lambda\) is a unit of order \(p-1\) in the ring of integers modulo \(p\) and \(G\) has class at most \(r(p-1)\), where \(r\geq 2\), then \(\phi^{p^{r-1}(p-1)}=1\).NEWLINENEWLINENEWLINELet \(R(2,5)=\langle x,y\rangle\). The author applies his result to the automorphism \(\phi\) of \(R\) given by \(\phi(x)=y\) and \(\phi(y)=xy\). Then using this he gets results about the sequence for various groups of exponent 5.