TITLE: Finite simple groups with upper bound on order of elements
QUESTION [8 upvotes]: Given a natural number k, are there only finitely many finite simple groups with the property that all elements have order at most k?
This holds if I only look at the finite simple groups I understand 
(e.g. alternating groups and SL(k,finite field)), but it's not clear to me whether this holds for all finite simple groups, even using  their classification.

REPLY [4 votes]: Indeed, a stronger claim is true: there are only finitely many finite simple groups $G$ such that the largest prime dividing $|G|$ (which is also the largest prime order of an element) is at most $k$.  For a proof, see Section 5 of:
L. Babai, A. J. Goodman and L. Pyber. Groups without faithful transitive permutation representations of small degree. J. Algebra 195 (1997), no. 1, 1–29.