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17.72 (2010)

Partially Solved

Let $AB$ be a Frobenius group with kernel $A$ and complement $B$. Suppose that $AB$ acts on a finite group $G$ so that $GA$ is also a Frobenius group with kernel $G$ and complement $A$.
$\qquad$ a) Is the nilpotency class of $G$ bounded in terms of $|B|$ and the class of $C_G(B)$?
$\qquad$ b) Is the exponent of $G$ bounded in terms of $|B|$ and the exponent of $C_G(B)$?

Contributor: V. D. Mazurov

Progress

a) Yes, it is (N. Yu. Makarenko, P. Shumyatsky, Proc. Amer. Math. Soc., 138 (2010), 3425–3436).

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