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19.75 (2018)

Solved

Let $P$ be a finite 2-group of exponent $2^e$ such that the rank of every abelian subgroup is at most $r$. Is it true that $|P| \leqslant 2^{r(e+1)}$? This bound would be sharp (for a direct product of quaternion groups).

Contributor: B. Sambale

Progress

No, it is not true, as follows from the examples constructed in (A. Yu. Olshanskii, Math. Notes, 23 (1978), 183–185) (A. Mann, Letter of 23 April 2018).

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