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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
CHEVIE
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