Infinite $p$-groups containing exactly $p^2$ solutions of the equation $x^p=1$

F. N. Liman

Sumy State Pedagogical Institute

Abstract: 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.

UDC: 519

Received: 17.09.1975

 English version: Mathematical Notes, 1976, 20:1, 563–567

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