Pure subgroup

In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas.

Definition

A subgroup $S$ of a (typically abelian) group $G$ is said to be pure if whenever an element of $S$ has an $n^\{th\}$ root in $G$, it necessarily has an $n^\{th\}$ root in $S$.

Origins

Pure subgroups are also called isolated subgroups or serving subgroups and were first investigated in Prüfer's 1923 paper [cite journal | title = Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen | first = H. | last = Prüfer | journal = Math. Zeit. | volume = 17 | pages=35–61 | year = 1923 | doi = 10.1007/BF01504333 | url = http://www.zentralblatt-math.org/zmath/en/search/?q=an:49.0084.03&format=complete | format = Dead link|date=May 2008 ] which described conditions for the decomposition of primary abelian groups as direct sums of cyclic groups using pure subgroups. The work of Prüfer was complimented nicely by Kulikoff [cite journal | title = Zur Theorie der Abelschen Gruppen von beliebiger Mächtigkeit | first = L. | last = Kulikoff | journal = Rec. Math. Moscou, n. Ser. | volume = 9 | pages = 165–181 | year = 1941 | url = http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0025.29901&format=complete | format = Dead link|date=May 2008 ] where many results are reproved using pure subgroups systematically. In particular, a nice proof is given that pure subgroups of finite exponent are direct summands. A more complete discussion of pure subgroups, their relation to infinite abelian group theory, and a survey of their literature is given in Irving Kaplansky's little red book [cite book | title = Infinite Abelian Groups | first = Irving | last = Kaplansky | author-link1 = Irving Kaplansky | publisher = University of Michigan | year = 1954 ] .

Examples

* Every direct summand of a group is a pure subgroup
* Every pure subgroup of a pure subgroup is pure.
* A divisible subgroup of an Abelian group is pure.
* If the quotient group is torsion-free, the subgroup is pure.
* The torsion subgroup of an Abelian group is pure.
* The union of pure subgroups is a pure subgroup.

Since in a finitely generated Abelian group the torsion subgroup is a direct summand, so one might wish the torsion subgroup was always a direct summand of an Abelian group. Unfortunately it is merely a limit and so is only a pure subgroup. Under certain mild conditions, pure subgroups are direct summands and so one can still recover the desired result under those conditions, as in Kulikoff's paper. This is a typical use of pure subgroups as an intermediate property between an original result on direct summands with finiteness conditions to a full result on direct summands with less restrictive finiteness conditions. Another example of this use is Prüfer's paper which takes the old result that finite torsion Abelian groups are direct sums of cyclic groups and through an intermediate consideration of pure subgroups shows that all torsion Abelian groups of finite exponent are direct sums of cyclic groups.

Generalizations

Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots. Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is a direct summand of a direct sum of finitely presented modules. In the case of the integers and Abelian groups, this simple means a direct sum of cyclic groups which was the original motivation for the study of pure subgroups in Prüfer's 1923 paper.

References

* Chapter III.