Abstract: Let G be a finite group and n be a natural number which is a product of distinct primes. Define
Hn (G) = < x 
G
| xn # 1>. In this work, we prove that the index
of Hn (G) in G must divide n under
the assumption that G is not a p- group where p is the set of all prime divisors of n. Moreover, we
determine the structure of G/Hn (G). This generalizes a result of Hughes and Thompson.
Furthermore, the structure of a finite solvable group H admitting an automorphism a of
prime order is studied and a bound for the Fitting length of H is obtained without the assumption
(0(a) , |H| ) = 1.