Geodesic
Partial generalization of one of Macdonald's results
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 8 (2011), pp. 369-371.

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It is proved that if in every finite subgroup of a $2$-group $G$ the identity $[x,y]^2=1$ holds, then this identity holds in $G$ also. In particular, $G$ is locally finite, its derived subgroup is of exponent 4, and the second derived subgroup belongs to the center of $G$.
Mots-clés : $p$-группа, коммутант. 
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D. V. Lytkina. Partial generalization of one of Macdonald's results. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 8 (2011), pp. 369-371. http://geodesic.mathdoc.fr/item/SEMR_2011_8_a30/

[1] I. D. Macdonald, “On certain varieties of groups”, Math. Z., 76:2 (1961), 270–282 | DOI | MR | Zbl

[2] GAP: Groups, algorithms and programming http://www.gap-system.org

[3] D. V. Lytkina, “On $2$-groups, all of whose finite subgroups are of nilpotency class $2$”, Sib. Electron. Math. Reports, 8 (2011), 1–3 | MR