In 1974, Vegh proved that if $k$ is a prime and $m$ a positive integer, there
is an $m$ term permutation chain of $k$th power residue for infinitely many
primes [E.Vegh, $k$th power residue chains, J.Number Theory, 9(1977), 179-181].
In fact, his proof showed that $1,2,2^2,...,2^{m-1}$ is an $m$ term permutation
chain of $k$th power residue for infinitely many primes. In this paper, we
prove that for any "possible" $m$ term sequence $r_1,r_2,...,r_m$, there are
infinitely many primes $p$ making it an $m$ term permutation chain of $k$th
power residue modulo $p$, where $k$ is an arbitrary positive integer [See
Theorem 1.2]. From our result, we see that Vegh's theorem holds for any
positive integer $k$, not only for prime numbers. In fact, we prove our result
in more generality where the integer ring $\Z$ is replaced by any $S$-integer
ring of global fields (i.e. algebraic number fields or algebraic function
fields over finite fields).