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It is a known fact that if a group $G$ has finitely many commutators then the order of the commutator subgroup is finite. I have found the following upper bound: Let the number of commutators be equal to $n<\infty$ then $$|G'|< n^{2n^2\log_2(n)}$$ I did this by combining the best bound given by a proof of Schur's theorem and the best bound given by a proof of the reverse Schur's theorem for finitely generated groups.

This upper bound does not seem very good. Are there better known bounds? For example by using a different method.

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    $\begingroup$ If $G$ has $n < \infty$ commutators, then each conjugacy class in $G$ is of size $\leq n$. Then Theorem 1.8 in R. M. Guralnick; A. Maróti; Average dimension of fixed point spaces with applications. Adv. Math. 226 (2011), no. 1, 298-308 gives $|G'| < n^{(1/2)(7 + \log_2 n)}$. $\endgroup$
    – spin
    Dec 1, 2022 at 15:26
  • $\begingroup$ Let $u_n$ be the largest cardinal of $|G'|$ for a group with exactly $n$ commutators, and $v_n=\max_{i\le n}u_i$. A (lengthy) verification shows, if I'm correct, that $u_n=n$ for $n\le 4$. Have further values been computed? $\endgroup$
    – YCor
    Dec 1, 2022 at 17:50
  • $\begingroup$ By the way, I'm not sure what is $n_0$, the smallest $n$ for which $u_n>n$. The examples in this answer show that for odd prime $p$ and $n\ge 4$ we have $v_{p^{2n}}>p^{n(n-1)/2}$; thus $n_0\le 3^8=6561$. This also yields $v_n\succeq n^2$. Examples of this flavor might suggest $v_n$ is indeed suprapolynomial, which would make the upper bound in spin's comment quite good. $\endgroup$
    – YCor
    Dec 1, 2022 at 17:53
  • $\begingroup$ For $n=5,6$ it is also the case that $u_n=n$. For higher values I don't know. The first $n$ for which I am aware that $u_n>n$ is 15. $\endgroup$
    – Lucas
    Dec 2, 2022 at 0:12
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    $\begingroup$ Theorem D in R. M. Guralnick; Commutators and commutator subgroups. Adv. in Math. 45 (1982), no. 3, 319-330 implies that if $|G'| < 16$, then $G'$ is equal to the set of commutators. So $n_0 = 15$. $\endgroup$
    – spin
    Dec 2, 2022 at 3:24

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