Derived Length and Fitting Height of Finite Simply Reducible Groups

ABSTRACT: The concept of simply reducible group is introduced by E. P. Wigner in his study of group representations and quantum mechanics in 1941. A finite group is called a simply reducible group if every element is conjugate to its inverse, and the tensor product of any two irreducible complex representations is multiplicity-free. In this paper, we work on the derived length and Fitting height of finite simply reducible groups. For each odd prime power $q$ and each integer $n \geq 2$, we construct a finite quaternion congruence quotient $G_{q,n}$ following the work of D. Prasad and C. Riehm. By direct calculations in these finite quotients, we prove that $G_{q,n}$ is a simply reducible group with derived length $\lceil\log_2(3n)\rceil$ and Fitting height $2$ or $3$. As a result, we see that the derived length is unbounded even for finite simply reducible groups of Fitting height $2$. This gives a negative answer to Kazarin's Problem 17.53(b) in the \textit{Kourovka Notebook}. Furthermore, we prove that every finite simply reducible group $G$ satisfies $G'\leq\Fit_4(G)$ and that $G/\Fit_4(G)$ is an elementary abelian $2$-group. In particular, we prove that the Fitting height satisfies $\fh(G)\leq5$, which gives an affirmative answer to Problem 17.53(a).