Vol. 36, No. 2, 1971

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A note on the minimality of certain bitransformation groups

Ta-Sun Wu

Vol. 36 (1971), No. 2, 553–556

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.

Mathematical Subject Classification
Primary: 54.80
Secondary: 22.00
Received: 15 April 1969
Published: 1 February 1971
Ta-Sun Wu