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:



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> 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) 
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(They may well be classical and well known results, hence the tag "reference request")



Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?
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. 


