Geodesic
On element orders in coverings of the simple groups $L_n(q)$ and $U_n(q)$
Trudy Instituta matematiki i mehaniki, Группы и графы, Tome 13 (2007) no. 1, pp. 89-98

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We prove that if a finite simple linear or unitary group defined over a field of characteristic $p$ and having dimension sufficiently large as compared with $p$ acts on a finite-dimensional vector space over some field of the same characteristic $p$, then the corresponding semidirect product contains an element whose order is distinct from any element order of the simple group. 
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A. V. Zavarnitsin; V. D. Mazurov. On element orders in coverings of the simple groups $L_n(q)$ and $U_n(q)$. Trudy Instituta matematiki i mehaniki, Группы и графы, Tome 13 (2007) no. 1, pp. 89-98. http://geodesic.mathdoc.fr/item/TIMM_2007_13_1_a5/