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Let $G$ be a finite group, $|G|=n$. Then $G$ embeds into $S_n$ via left multiplication.

I have just learned here that, if $\exists H \lhd G$ (proper normal subgroup) which has trivial centralizer in $G$, then $G$ embeds into $S_{|H|<n}$ by conjugacy (e.g., $G=S_n$ and $H=A_n$, for $n \ge 3$).

This kind of "sharpening process" made me ask the following: Can we always/in some case determine the least $l$ such that $G \hookrightarrow S_l$?

  • It's a duplicate, but the answer to the question "Is it possible to determine the least such $l$" is clearly yes. I am not sure exactly what you mean by "Can we determine ..." (and there is more than one reasonable interpretation). – Derek Holt Nov 15 '19 at 13:52
  • Unfortunately, to non-English natives "can we" and "is it possible" sound the same, which evidently it is not. –  Nov 15 '19 at 14:01
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    The instance I had in mind was a little obscure, and had nothing to do with non-English speakers. The authors (who were English speakers) wrote "we can determine X" in the theorem statement and what they proved was that there exists an algorithm to determine X, but their proof of the existence of the algorithm was non-constructive, so it was not clear whether they really could determine X. But for your question, it is easy to write down an algorithm to solve this problem. But there is no known polynomial-time algorithm, and it is suspected that there is none such. So the problem is hard. – Derek Holt Nov 15 '19 at 14:08
  • Thanks. I'll go through the linked post to learn more. –  Nov 15 '19 at 14:14

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