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Consider the map $\zeta: \{ \mbox{division rings} \} \mapsto \{ \mbox{groups} \}: k \mapsto \mathbf{PGL}_2(k)$.

Is this map known to be an injection - in other words, if $k$ and $k'$ are nonisomorphic division rings, are the images $\mathbf{PGL}_2(k)$ and $\mathbf{PGL}_2(k')$ also nonisomorphic ?

Does the same hold in the other dimensions ?

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    $\begingroup$ I'd have distinguished between two different kinds of arrows, thus: $\zeta: \{ \mbox{division rings} \} \to \{ \mbox{groups} \}: k \mapsto \mathbf{PGL}_2(k).$ Do you disagree with that? $\endgroup$
    – Michael Hardy
    Commented Oct 12, 2022 at 3:36

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PSL remembers everything!

What's written above is true for all $n, n_1 \geq 2$ and all skew fields except two finite cases: $PSL(2, \Bbb F_7) \cong PSL(3, \Bbb F_2)$, $PSL(2, \Bbb F_4) \cong PSL(2, \Bbb F_5)$. (V. M. Petechuk, Isomorphisms between linear groups over division rings, Can. J. Math.Vol. 45 (5), 1993 pp. 997-1008). Case $n \geq 3$ is more or less folklore, with field case settled by Shreier-wan der Waerden, and skew field case by Dieudonne-Hua.

Addressing exact question on $PGL(n)$: I remember convincing myself a few years ago that isomorphism between $PGL(2)$-s over infinite skew fields should induce two-way inclusions between according $PSL(2)$-s, and a posteriori isomorphism, so claim should follow from $PSL$ reflecting isomorphisms; unfortunately, I have forgot exact arguments and have failed to reproduce them now. For $n \geq 3$ you can find exact proof in Hahn's paper referenced below, but dimension restriction is unavoidable if you follow his strategy.

Sligtly more general case where $D$ is merely a subring of division ring (but $n \geq 5$) can be found in O.T. O'Meara, A general isomorphism theory for linear groups, 1977. Possibilities of producing Morita equivalences from abstract group isomorphisms (for $n \geq 3$) is covered in A. J. Hahn, Category equivalences and linear groups over rings. Journal of Algebra, 77(2), 505–543.

As far as I know, possibility of abstract isomorphism between different linear groups over non-isomorphic skew fields is still an open problem.

An outdated, but pretty comprehensive survey on related topics can be found here https://web.osu.cz/~Zusmanovich/links/files/weisfeiler/1981-commalg.pdf

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  • $\begingroup$ In O'Meara's paper, the author seems to be assuming that $n \geq 5$ ? Are the cases $n = 2, 3, 4$ also covered (somewhere) ? $\endgroup$
    – THC
    Commented Oct 11, 2022 at 14:05
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    $\begingroup$ @THC ...Well, I was a bit too eager to include $n = 2$ case without some nuances; I'll edit the answer to promote it from containing almost true statements to being true. $\endgroup$
    – Denis T
    Commented Oct 11, 2022 at 14:52
  • $\begingroup$ Doesn't the case $n = 2$ (for $\mathbf{PGL}_2(D)$) directly follows from the case of $\mathbf{PSL}_2(D)$ plus the fact that the latter is generically a characteristic subgroup of $\mathbf{PGL}_2(D)$ ? (An isomorphism between the $\mathbf{PGL}_2$s induces an isomorphism between the $\mathbf{PSL}_2$s.) $\endgroup$
    – THC
    Commented Oct 11, 2022 at 17:44

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