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Trivial intersection of blocks and nilpotent subgroups. (English) Zbl 07209589
J. Algebra 559, 510-528 (2020); addendum ibid. 584, 161-162 (2021).
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 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.

MSC:
20C20 Modular representations and characters
20C15 Ordinary representations and characters
20C33 Representations of finite groups of Lie type
20C30 Representations of finite symmetric groups
Software:
CHEVIE
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References:
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