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let $G$ be a group. Is it true that the only subgroup of $G$ that is conjugate to $G$ is $G$ itself? if $G$ is finite this is clear as conjugate subgroups have the same order but what about infinite groups?

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    @Mark Bennet: it is $xyx^{-1}$ that is in the image $H$ not $y$2011-07-02

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Each conjugation induces a permutation of $G$ as a set. When you permute a set, proper subsets remain proper subsets.

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    thanks alot!! it's clea$r$ now2011-07-02