Pure subgroup
From Encyclopedia of Mathematics
serving subgroup
A subgroup 
of an Abelian group 
such that for any 
the solvability of the equation 
in 
implies its solvability in 
. Examples of pure subgroups are the zero subgroup, 
itself, the torsion part of 
, and direct summands. Not every pure subgroup need be a direct summand, even for a 
-group. However, if 
is a torsion pure subgroup of an Abelian group 
and if the orders of its elements are uniformly bounded, then 
is a direct summand in 
. There is a complete description of the Abelian groups in which every pure subgroup is a direct summand (see [1]). The question of the cardinality of the set of pure subgroups of an Abelian group has been thoroughly investigated.
References
| [1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
Comments
References
| [a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1982) |
How to Cite This Entry:
Pure subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pure_subgroup&oldid=16485
Pure subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pure_subgroup&oldid=16485
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article