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Let (T,X) be a transformation
group with compact Hausdorff space X and topological group T. Let (X,G) be a
transformation group with G a compact topological group. Then the triple
(T,X,G) is a bitransformation group if (tx)g = t(xg) for all t ∈ T,x ∈ X,g ∈ G
and the action of G on X is strongly effective, (that is xg = x if and only
if g = the identity element e of G). A bitransformation group (T,X,G),
induces canonically the transformation group (T,X∕G) where X∕G is the
orbit space of (X,G). Let (T,X,G) be a bitransformation group. Suppose
(T,X∕G) is a minimal transformation group whereas (T,X) is not a minimal
transformation group then what is the possible structure of (T,X,G)? In this note, it
is proved that the fundamental group of X must be of certain form when
G is a circle group. Use this result together with some results of Malcev,
a necessary and sufficient condition is found for the minimality of certain
nilflows.
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