an:07209589
Zbl 7209589
Liu, Yanjun; Willems, Wolfgang; Xiong, Huan; Zhang, Jiping
Trivial intersection of blocks and nilpotent subgroups
EN
J. Algebra 559, 510-528 (2020); addendum ibid. 584, 161-162 (2021).
0021-8693
2020
j
20C20 20C15 20C33 20C30
principal block; intersection of blocks; Hall subgroups; generalized \(p^\prime \)-core
Summary: Let \(p, q\) be different primes and suppose that the principal \(p\)- and the principal \(q\)-block of a finite group have only one irreducible complex character in common, namely the trivial one. We conjecture that this condition implies the existence of a nilpotent Hall \(\{p, q \}\)-subgroup and prove that a minimal counter-example must be an almost simple group where \textit{pq} divides the order of its simple nonabelian normal subgroup. As an immediate consequence we obtain that the conjecture holds true for \(p\)-solvable or \(q\)-solvable groups. Furthermore, we prove the conjecture in case \(2 \in \{p, q \}\) using the classification theorem of finite simple groups. Finally, we consider the situation that the intersection of an arbitrary \(p\)-block with an arbitrary \(q\)-block contains only one irreducible character.