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11.11 (1990)

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The well-known Baer-Suzuki theorem states that if every two conjugates of an element $a$ of a finite group $G$ generate a finite $p$-subgroup, then $a$ is contained in a normal $p$-subgroup.
$\qquad$ a) Does such a theorem hold in the class of periodic groups? The case $p = 2$ is of particular interest.
$\qquad$ b) Does such a theorem hold in the class of binary finite groups?

Contributor: A. V. Borovik

Progress

a) No, it does not hold for $p = 2$ (V. D. Mazurov, A. Yu. Olshanskii, A. I. Sozutov, Algebra and Logic, 54, no. 2 (2015), 161–166).
b) Yes, such a theorem does hold for binary finite groups. By the Baer–Suzuki theorem $\langle a_1, b \rangle$ is a finite $p$-group for any element $a_1 \in a^G = \{a^g \mid g \in G\}$ and for any $p$-element $b \in G$. Now induction on $n$ yields that the product $a_1 \dots a_n$ is a $p$-element for any $a_1, \dots, a_n \in a^G$. (A. I. Sozutov, Siberian Math. J., 41, no. 3 (2000), 561–562.)

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