Geodesic
An infinite group with subgroups of prime orders
Izvestiya. Mathematics, Tome 16 (1981) no. 2, pp. 279-289
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An infinite nonabelian group all of whose proper subgroups have prime order is constructed. This solves a problem of O. Yu. Schmidt. The proof involves modifying and extending the series of lemmas in an earlier paper of the author. Bibliography: 3 titles. 
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A. Yu. Ol'shanskii. An infinite group with subgroups of prime orders. Izvestiya. Mathematics, Tome 16 (1981) no. 2, pp. 279-289. http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a3/

[1] Olshanskii A. Yu., “Beskonechnye gruppy s tsiklicheskimi podgruppami”, Dokl. AN SSSR, 245:4 (1979), 785–787 | MR

[2] Olshanskii A. Yu., “Beskonechnaya prostaya neterova gruppa bez krucheniya”, Izv. AN SSSR. Ser. matem., 43 (1979), 1328–1393 | MR

[3] Adyan S. I., Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl