Fibonacci lengths of all finite $p$-groups of exponent $p^2$

Abstract

The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent $p$ and the lengths depend on the Wall number $k(p)$. The p-groups of nilpotency class 3 and exponent $p$ were studied in 2004 also by Dikici. In the paper, we study all $p$-groups of nilpotency class 3 and exponent $p^2$. Thus, we complete the study of Fibonacci lengths of all $p$-groups of order $p^4$ by proving that the Fibonacci length is $k(p^2)$.