View source for Divisible group

A group in which for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336601.png" /> and for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336602.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336603.png" /> is solvable. The group is usually understood to be Abelian. Important examples of divisible groups are the additive group of all rational numbers and the group of all complex roots of unity of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336605.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336606.png" /> is a prime number (the quasi-cyclic group). Any Abelian divisible group splits into a direct sum of groups each of which is isomorphic to one of the groups mentioned in the examples. Much less is known about non-Abelian divisible groups (also called complete groups). Any divisible group, except the identity group, is infinite. Any group is imbeddable in a suitable divisible group. If the equations stated in the definition of a divisible group have a unique solution, the group is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336608.png" />-group. Examples, in particular, are locally nilpotent divisible torsion-free groups.

====References====
<table><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></table>



====Comments====
An [[Abelian group|Abelian group]] is divisible if and only if, regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d0336609.png" />-module, it is injective (cf. [[Injective module|Injective module]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366010.png" /> be the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366011.png" />-adic numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366012.png" /> its ring of integers. Then the quasi-cyclic group for the prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366013.png" /> is the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366014.png" /> which is also the injective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366015.png" /> for the imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033660/d03366017.png" />.

See also [[Split group|Split group]]; [[Splittable group|Splittable group]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,   "Infinite abelian groups" , '''1''' , Acad. Press  (1970)  pp. Chapt. 4</TD></TR></table>