- Definition
- Examples
- Properties
- References
In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below. Definition Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with termwise addition. For each n in N, let en be the sequence with n-th term equal to 1 and all other terms 0. A torsion-free abelian group G is said to be slender if every homomorphism from ZN into G maps all but finitely many of the en to the identity element. Examples Every free abelian group is slender. The additive group of rational numbers Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced. Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender. Properties - A torsion-free abelian group is slender if and only if it is reduced and contains no copy of the Baer–Specker group and no copy of the p-adic integers for any p.
- Direct sums of slender groups are also slender.
- Subgroups of slender groups are slender.
- Every homomorphism from ZN into a slender group factors through Zn for some natural number n.
References - {{Cite book | last=Fuchs | first=László | title=Infinite abelian groups. Vol. II | publisher=Academic Press | location=Boston, MA | zbl=0257.20035 | mr=0349869 | year=1973 | at=Chapter XIII | series=Pure and Applied Mathematics | volume=36 }}.
- {{cite book | first=Phillip A. | last=Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | zbl=0204.35001 | pages=111–112}}
- {{cite journal | first=R. J. | last=Nunke | title=Slender groups | journal=Bulletin of the American Mathematical Society | volume=67 | year=1961 | issue=3 | pages=274–275 | doi=10.1090/S0002-9904-1961-10582-X | zbl=0099.01301 }}
- {{cite journal | author1-link=Saharon Shelah | first1=Saharon | last1=Shelah | first2=Oren | last2=Kolman | title=Infinitary axiomatizability of slender and cotorsion-free groups | journal=Bulletin of the Belgian Mathematical Society | volume=7 | year=2000 | mr=1806941 | zbl=0974.03036 | url=http://projecteuclid.org/euclid.bbms/1103055621 | pages=623–629 }}
{{Abstract-algebra-stub}} 2 : Properties of groups|Abelian group theory |