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Is the following statement true or not?

A locally compact Hausdorff space $X$ is a group if and only if its Stone–Čech compactification$\beta X$ is a group.

Thanks.

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    Here a group means that it is a topological group under the given topology.2012-12-23

1 Answers 1

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I assume both occurrences of "is a group" mean "is the underlying space of a topological group." Then Henno Brandsma's first comment gives a counterexample, because $\beta\mathbb Z$ is indeed not the underlying space of a topological group. The reason is that it is not homogeneous: The points in $\mathbb Z$ are isolated and the others are not.

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    Thanks for answering my question. Bohr compactification is just for groups. I am wondering that given a locally compact Hausdorff space $X$, is there any compactification of $X$, say $Y$, such that $X$ is a topological group if and only if $Y$ is a compact group?2012-12-23