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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-Cech compactification $\beta X$ is a group.

Thanks.

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$\beta\mathbb{Z}$ is not a group, I believe, while $\mathbb{Z}$ is... –  Henno Brandsma Dec 22 '12 at 21:28
    
In general (see dutiaw37.twi.tudelft.nl/~kp/onderwijs/topologie/d17-betaX.pdf, e.g.) we have that a pseudocompact (Tychonoff) topological group is such that its group operations can be extended to its Cech-Stone compactification. –  Henno Brandsma Dec 22 '12 at 21:32
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You should explain what "is a group" means. –  Chris Eagle Dec 22 '12 at 22:42
    
There is a way to compactify topological groups (en.wikipedia.org/wiki/Bohr_compactification) but the underlying topological space of the Bohr compactification isn't the Stone-Cech compactification. For example, the Bohr compactification of $\mathbb{Z}$ is the Pontrjagin dual of $S^1$ with the discrete topology... –  Qiaochu Yuan Dec 22 '12 at 23:08
    
Here a group means that it is a topological group under the given topology. –  user16283 Dec 23 '12 at 19:31

1 Answer 1

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? –  user16283 Dec 23 '12 at 19:36

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