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21.51 (2026)

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Let $p$ be a prime, and $P$ a finite $p$-group.
$\qquad$ (a) Suppose that $P$ has an abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal abelian subgroup of order $p^n$?
$\qquad$ (b) Suppose that $P$ has an elementary abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal elementary abelian subgroup of order $p^n$?

Contributor: G. Glauberman

Progress

It is easy to see that for $p = 2$, the answer to (b) is “yes” only for $n = 1$. The answer to both questions is “yes” for $n < (p + 2)/2$ (G. G. Glauberman, J. Algebra, 319, no. 2 (2008), 800–805), as well as for $n \leqslant 5$ when $p \neq 2$ (M. Konvisser, D. Jonah, J. Algebra, 34 (1975), 309–330). The answer to both questions is “no” for $n \geqslant (p + 9)/2$ when $p \geqslant 5$ (for $p \geqslant 7$ due to G. Glauberman, Contemp. Math., 524 (2010), 61–65; for $p = 3, 5$ due to Ya. G. Berkovich, J. Algebra, 248, no. 2 (2002), 472–553). Thus, the only open cases for $p \geqslant 5$ are $n = 6$ for $p = 5$ , and $n = (p + 3)/2, (p + 5)/2, (p + 7)/2$ for $p > 5$.

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