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Functorial radicals and non-abelian torsion, II
Published online by Cambridge University Press: 20 January 2009
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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.
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- 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), 317–329.CrossRefGoogle Scholar
2.Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory (Interscience, NY and London, 1966).Google Scholar
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