I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
1) Are there two finite subgroups $P,P'\subset\mathrm{GL}(n,\mathbb{Z})$ that are abstractly isomorphic but not conjugate in $\mathrm{GL}(n,\mathbb{Z})$?
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2) Are there two finite subgroups $P,P'\subset\mathrm{GL}(n,\mathbb{Z})$ that are abstractly isomorphic but not conjugate in $\mathrm{GL}(n,\mathbb{R})$?
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3) What if we don't assume $P$ and $P'$ to be finite? (ok, this has nothing to do with crystallography)
(They may well be classical and well known results, hence the tag "reference request")
Best Answer
The answer to all three questions is yes and certainly is classical.
One simple example is the following:
Let $C_2$ act faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivial element of $C_2$ swaps 1,2 and also swaps 3,4. In the second the non-trivial element swaps 1,2 and fixes 3,4.
Each action defines a representation of $C_2$ on $\mathbb{Z}^4$ via permuation matrices. In one case the trace of the non-trivial permutation matrix is $0$ in the other $2$ so the images cannot be conjugate in $GL_4(\mathbb{Z})$ or $GL_4(\mathbb{R})$ however they both generate a subgroup $C_2$ in the former.
It is fairly clear this idea generalises to any isomorphism class of finite groups.