View source for Locally nilpotent group

A group in which every finitely-generated subgroup is nilpotent (see [[Nilpotent group]]; [[Finitely-generated group]]). In a locally nilpotent group the elements of finite order form a [[normal subgroup]], the torsion part of this group (cf. [[Periodic group]]). This subgroup is the direct product of its [[Sylow subgroup]]s, and the quotient group with respect to it is torsion-free. A locally nilpotent torsion-free group (cf. [[Group without torsion]]) has the uniqueness-of-roots property: If, for elements $a$ and $b$ and any integer $n\ne0$, one has $a^n = b^n$, then $a=b$. Every locally nilpotent torsion-free group $G$ has a Mal'tsev completion, that is, it can be imbedded in a unique locally nilpotent torsion-free group $G^*$ such that all equations of the form $x^n=g$ are solvable in $G^*$, where $n \ne 0$ and $g$ is any element of $G$. This completion is functorial, that is, any homomorphism $f : G_1 \rightarrow G_2$ of locally nilpotent torsion-free groups $G_1$ and $G_2$ can be uniquely extended to a homomorphism $f^* : G_1^* \rightarrow G_2^*$.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR>
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