Geodesic
A~finitely generated complete group
Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 233-277

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An example of a $2$-generated complete group (different from the identity) with unique extraction of roots is constructed. An example is indicated, in passing, of a noncyclic $2$-generated group in which every element is conjugate to some power of a fixed element. It is proved that there are continuum many nonisomorphic such examples. The proof is based on a method developed in several papers of A. Yu. Ol'shanskii. Bibliography: 8 titles. 
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     author = {V. S. Guba},
     title = {A~finitely generated complete group},
     journal = {Izvestiya. Mathematics },
     pages = {233--277},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a0/}
}
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V. S. Guba. A~finitely generated complete group. Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 233-277. http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a0/