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Let (X,T,Π) be a topological
transformation group, where X is a nontrivial Hausdorff continuum, and T is a
topological group which leaves an endpoint e of X fixed. Wallace showed that if X is
locally connected and T is cyclic, T has another fixed point. In a later paper, Wallace
asked the following question: if X is a peano continuum and T is compact or abelian,
does T have another fixed point?
In 1952, Wang showed that if X is arcwise connected and T is compact, T has
another fixed point; Chu has recently extended this result by showing T has infinitely
many fixed points. Gray has shown that in the abelian case, the answer to Wallace’s
question is “no” (in general). However, if T is a generative group, and if X is arcwise
connected, T has another fixed point. In this paper we will generalize the last result.
In fact, we show that if X is arcwise connected or locally connected, and T is a group
of the form AH, where H is a connected subgroup, and A is an abelian group
generated by a compact subset, and A lies in the center of T, then T has another
fixed point. We will generalize several known theorems by studying ordered
spaces similar to those introduced by Wallace in 1945; in particular, we
will obtain a generalized solution of the compact group problem (Theorem
2).
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