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16.33 (2006)

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Suppose that a finite $p$-group $G$ has an abelian subgroup $A$ of order $p^n$. Does $G$ contain an abelian subgroup $B$ of order $p^n$ that is normal in $\langle B^G \rangle$
$\qquad$ a) if $p = 3$?
$\qquad$ b) if $p = 2$?
$\qquad$ c) If $p = 3$ and $A$ is elementary abelian, does $G$ contain an elementary abelian subgroup $B$ of order $3^n$ that is normal in $\langle B^G \rangle$?

Contributor: G. Glauberman

Progress

The corresponding results have been proved for greater primes $p$, based on extensions of Thompson’s Replacement Theorem (and the dihedral group of order 32 shows that the third question has negative answer for $p = 2$). See (G. Glauberman, J. Algebra, 196 (1997), 301–338; J. Alperin, G. Glauberman, J. Algebra, 203 (1998), 533–566; G. Glauberman, J. Algebra, 272 (2004), 128–153).

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