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Recursively presented Abelian groups: Effective p-Group theory. I

Published online by Cambridge University Press:  12 March 2014

Charlotte Lin
Charlotte Lin*
Affiliation:
Schlumberger-Doll Research, Ridgefield, Connecticut 06877
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Extract

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.

We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)

The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, proving

Theorem. Every group of bounded order is a direct sum of cyclic groups, and

Theorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 617 - 624
Copyright
Copyright © Association for Symbolic Logic 1981

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References

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