Let H be a finite group, and let θ be an automorphism of H whose order divides n. Hall proved that the number of elements x of H that satisfy the relation x*xθ*xθ2 ***xθn-1 = 1 is a multiple of gcd(n,|H|). For a prime factor p of gcd(n,|H|), if such a number is not a multiple of gcd(pn,|H|), then a Sylow p-subgroup of H is exceptional.
Hall's Relations in Finite Groups
JA_271_1_312_326
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