Derived Length and Fitting Height of Finite Simply Reducible Groups

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 ≥ 2, we construct a finite quaternion congruence quotient Gq,n following the work of D. Prasad and C. Riehm. By direct calculations in these finite quotients, we prove that Gq,n is a simply reducible group with derived length log₂(3n) 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 Kourovka Notebook. Furthermore, we prove that every finite simply reducible group G satisfies G'≤₄(G) and that G/₄(G) is an elementary abelian $2$-group. In particular, we prove that the Fitting height satisfies (G)≤5, which gives an affirmative answer to Problem 17.53(a).