Geodesic
Infinite $p$-groups containing exactly $p^2$ solutions of the equation $x^p=1$
Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 11-18 
F. N. Liman. Infinite $p$-groups containing exactly $p^2$ solutions of the equation $x^p=1$. Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 11-18. http://geodesic.mathdoc.fr/item/MZM_1976_20_1_a1/
@article{MZM_1976_20_1_a1,
     author = {F. N. Liman},
     title = {Infinite $p$-groups containing exactly $p^2$ solutions of the equation $x^p=1$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {11--18},
     year = {1976},
     volume = {20},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_1_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study arbitrary infinite 2-groups with three involutions and infinite locally finite $p$-groups ($p\ne2$), containing $p^2-1$ elements of order $p$. For odd $p$ the group $G=A\langle b\rangle$, where $A$ is a direct product of two quasicyclic 3-groups $|b|=9$, $b^3\in A$, and subgroup $A$ is generated by the elements of the commutator ladder of element $b$, is a unique infinite non-Abelian locally finite $p$-group whose equation $x^p=1$ has $p^2$ solutions.