  A group G is almost cyclic if there is an element x in G, such that for all g
in G, there is an element y in G and an integer n with ygy^{-1} = x^n (that is,
every element is conjugate to some power of x). W. Ziller asked whether there
are finitely-presented almost cyclic groups which are not cyclic in connection
with work on closed geodesics. V. Guba constructed an infinite (non-cyclic)
finitely generated almost cyclic group. The principal results of this paper
are: Solvable almost cyclic groups are cyclic, and one-relator almost cyclic
groups are cyclic.
