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In this note we present a simplified and slightly generalized version of alemma the authors published in 1987. The lemma as stated here asserts that ifthe order of a permutation of $n$ elements is greater than $n^{\alpha}$ thensome non-identity power of the permutation has support size less than$n/{\alpha}$. The original version made an unnecessary additional assumption onthe cycle structure of the permutation; the proof of the present cleanerversion follows the original proof verbatim. Application areas include paralleland sequential algorithms for permutation groups, the diameter of Cayley graphsof permutation groups, and the automorphisms of structures with regularityconstraints such as Latin squares, Steiner 2-designs, and strongly regulargraphs. This note also serves as a modest tribute to the junior author whoseuntimely passing is deeply mourned.

Element order versus minimal degree in permutation groups: an old lemma with new applications