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Functorial radicals and non-abelian torsion, II

Published online by Cambridge University Press:  20 January 2009

Shalom Feigelstock
Affiliation:
Bar-Ilan University, Ramat-Gan, Israel
Aaron Klein
Affiliation:
Bar-Ilan University, Ramat-Gan, Israel
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The object of this paper is to complete and continue some matters in [1].

In [1], Section 2, the torsion and torsion-free functors, whose operation on the category of abelian groups are well known, were extended to the category of all groups as follows. For a group A, put t0(A)= 0 and t1(A) = the subgroup of A generated by the torsion elements of A. Inductively define tn+1(A)/tn(A)=t1(A)/tn(A)), for every positive integer n. Then T(A)=∪ntn(A) is the smallest subgroup H of A such that A/H is torsion-free, [1], Th. 2.2. A group A satisfying T(A) = A was called a pre-torsion group. In [1], 2.12 an example was constructed of a group A satisfying t1(A)≠t2(A)=A. The question was posed whether for every positive integer n there exist groups A, satisfying tn–1(A)≠tn(A)=A. Here we give an affirmative answer. In fact, such groups will be constructed, as well as pre-torsion groups A with tk(A)≠A for every positive integer k, see Section 1.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Feigelstock, S. and Klein, A., Functorial radicals and non-abelian torsion, Proc. Edinburgh Math. Soc. 23 (1980), 317329.CrossRefGoogle Scholar
2.Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory (Interscience, NY and London, 1966).Google Scholar
3.Neumann, H., Varieties of Groups (Springer, NY, 1967).CrossRefGoogle Scholar