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14.60 (1999)

Solved

Suppose that $H$ is a non-trivial normal subgroup of a finite group $G$ such that the factor-group $G/H$ is isomorphic to one of the simple groups $L_n(q)$, $n \geqslant 3$. Is it true that $G$ has an element whose order is distinct from the order of any element in $G/H$?

Contributor: V. D. Mazurov

Progress

Yes, it is true: for $n \neq 4$ proved in (A. V. Zavarnitsine, Siberian Math. J., 49 (2008), 246–256), and for $n = 4$ in (M. A. Grechkoseeva, S. V. Skresanov, Siberian Electron. Math. Rep., 17 (2020), 585–589). Editors' comment: previous claim that it is not true for $n=4$ was erroneous.

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