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For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on.

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  • $\begingroup$ I believe this MSE question helps with the (finite) group case. math.stackexchange.com/questions/1599089/… $\endgroup$
    – walkar
    Commented Sep 26, 2023 at 14:23
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    $\begingroup$ This question for groups is a difficult question and has lots of history. Look for minimum faithful degree. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 26, 2023 at 14:33
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    $\begingroup$ You can start with this paper D. L. Johnson. Minimal permutation representations of finite groups. Amer. J. Math., 93:857–866, 1971. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 26, 2023 at 14:35
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    $\begingroup$ And here is a more recent reference . A. Grechkoseeva. On minimal permutation representations of classical simple groups. Sibirsk. Mat. Zh., 44(3):560–586, 2003. $\endgroup$
    – Benjamin Steinberg
    Commented Sep 26, 2023 at 14:36

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