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11.17 (1990)

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Let $G$ be a finite group and let $d = d(G)$ be the least positive integer such that $G$ satisfies a law $[x, \phantom{}_r y] = [x, \phantom{}_{r+d} y]$ (see 11.16) for some nonnegative integer $r = r(G)$.
$\qquad$ a) Let $e = 1$ if $d(G)$ is even and $e = 2$ otherwise. Is it true that the exponent of $G/F(G)$ divides $e \cdot d(G)$?
$\qquad$ b) If $G$ is a nonabelian simple group, does the exponent of $G$ divide $d(G)$?

Contributor: R. Brandl

Progress

Comments from the Notebook: Part a) is true for soluble groups (N. D. Gupta, H. Heineken, Math. Z., 95 (1967), 276–287). I have checked part b) for $\mathfrak{A}_n$, $PSL(2, q)$, and a number of sporadic groups.

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