The Glauberman Correspondence and Subgroups of Operator Groups: a Counterexample

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let π( G , A ):Irr A ( G )→Irr(C G ( A )) be the Glauberman–Isaacs correspondence. Let B ⩽ A and χ∈Irr A ( G ). We exhibit a counterexample to the conjecture that χπ( G , A ) is an irreducible constituent of the restriction of χπ( G , B ) to C G ( A ). 1991 Mathematics Subject Classification 20C15.