Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order \(4\) (Q1891707)

Let \(b\) be a block of a finite group \(G\) with abelian defect group \(P\). A conjecture by M. Broué says that \(b\) is isotypic to its Brauer correspondent \(c\) in \(N_G (P)\). The authors verify this conjecture in the special case where the inertial quotient \(N_G(P, e)/PC_G(P)\) is a cyclic group of order 4; here \(e\) denotes a block of \(C_G(P)\) such that \(e^G = b\). Combined with earlier results this implies that Broué's conjecture holds whenever \(|N_G (P,e)/PC_G(P)|\leq 4\). It follows that in these cases there exists a perfect isometry between \(b\) and \(c\); in particular, \(b\) and \(c\) have the same numbers of irreducible ordinary and modular characters, and all irreducible characters in \(b\) have height zero.