Geodesic
The formula of maximal possible rank of commutator subgroups of finite $p$-groups
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 804-808.

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All groups in the abstract are finite. We define rank $d(G)$ of a $p$-group $G$ as the minimal number of generators of $G$. In this paper, we obtain a compact formula for the strict upper bound of the ranks of commutator subgroups of finite $p$-groups generated by elements of given orders. This bound was described in a recent article of the author. But the corresponding formula was very complicated although containing some useful information. The new formula is much more simple and clear.
Keywords: finite $p$-group generated by elements of orders $p^{k_1},\dots,p^{k_n}$, number of generators of commutator subgroup of a finite $p$-group. 
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B. M. Veretennikov. The formula of maximal possible rank of commutator subgroups of finite $p$-groups. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 2, pp. 804-808. http://geodesic.mathdoc.fr/item/SEMR_2022_19_2_a7/

[1] B.M. Veretennikov, “The strict upper bound of ranks of commutator subgroups of finite $p$-groups”, Sib. Èlectron. Mat. Izv., 16 (2019), 1885–1900 | DOI | MR | Zbl

[2] B.M. Veretennikov, “On the commutator subgroups of finite 2-groups generated by involutions”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 77–84 | DOI | MR