{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:57:43Z","timestamp":1775462263443,"version":"3.50.1"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11880,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1981,9]]},"abstract":"<jats:p>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\u2032cev (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.<\/jats:p><jats:p>We investigate the effective content of the structure theory of <jats:italic>p<\/jats:italic>-groups. Recall that a <jats:italic>p<\/jats:italic>-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime <jats:italic>p<\/jats:italic>. (In the sequel, \u201cgroup\u201d = \u201cadditively written abelian group\u201d.)<\/jats:p><jats:p>The structure theory of <jats:italic>p<\/jats:italic>-groups is based on the two elementary notions of order and height. Recall that the <jats:italic>order of x<\/jats:italic> is the least integer <jats:italic>n<\/jats:italic> such that <jats:italic>nx<\/jats:italic> = 0. The <jats:italic>height of x<\/jats:italic> is the number of times <jats:italic>p<\/jats:italic> divides <jats:italic>x<\/jats:italic>, that is, the least <jats:italic>n<\/jats:italic> such that <jats:italic>x<\/jats:italic> = <jats:italic>p<jats:sup>n<\/jats:sup>y<\/jats:italic> for some <jats:italic>y<\/jats:italic> in the group but <jats:italic>x<\/jats:italic> \u2260 <jats:italic>p<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic>+1<\/jats:sup><jats:italic>y<\/jats:italic> for any <jats:italic>y<\/jats:italic>. If for each <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic> there is a \u201c<jats:italic>p<jats:sup>n<\/jats:sup><\/jats:italic>th-root\u201d <jats:italic>y<jats:sub>n<\/jats:sub><\/jats:italic>, so that <jats:italic>x<\/jats:italic> = <jats:italic>p<jats:sup>n<\/jats:sup>y<jats:sub>n<\/jats:sub><\/jats:italic>, then we say that <jats:italic>x<\/jats:italic> has <jats:italic>infinite height<\/jats:italic>. In 1923, Pr\u00fcfer related the two notions as criteria for direct sum decomposition, proving<\/jats:p><jats:p>Theorem. <jats:italic>Every group of bounded order is a direct sum of cyclic groups, and<\/jats:italic><\/jats:p><jats:p>Theorem. <jats:italic>Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273759","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:56:46Z","timestamp":1146938206000},"page":"617-624","source":"Crossref","is-referenced-by-count":6,"title":["Recursively presented Abelian groups: Effective <i>p<\/i>-Group theory. I"],"prefix":"10.1017","volume":"46","author":[{"given":"Charlotte","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200045436_ref018","unstructured":"Millar T. , Foundations of recursive model theory, this Journal (to appear)."},{"key":"S0022481200045436_ref025","unstructured":"Smith R. , The theory of profinite groups with effective presentations, Ph.D. dissertation, Pennsylvania State University, University Park, 1979."},{"key":"S0022481200045436_ref021","first-page":"341","article-title":"Computable algebra, general theory and theory of computable fields","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200045436_ref002","unstructured":"Crossley J. N. and Nerode A. , Effective dimension, Journal of Algebra (to appear)."},{"key":"S0022481200045436_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90006-9"},{"key":"S0022481200045436_ref023","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1973.45.621"},{"key":"S0022481200045436_ref010","first-page":"481","volume":"42","author":"Kalantari","year":"1977","journal-title":"Maximal vector spaces under automorphisms of the lattice of recursively enumerable sets"},{"key":"S0022481200045436_ref004","first-page":"477","volume":"36","author":"Dekker","year":"1971","journal-title":"Countable vector spaces with recursive operations. Part II"},{"key":"S0022481200045436_ref013","first-page":"129","article-title":"K teorii abelevych grupp proizvolnoi mo\u0161\u010dnosti","volume":"16","author":"Kulikov","year":"1945","journal-title":"Matemati\u010deski\u01d0 Sbornik"},{"key":"S0022481200045436_ref024","first-page":"13","volume":"43","author":"Shore","year":"1978","journal-title":"Controlling the dependence degree of a recursively enumerable vector space"},{"key":"S0022481200045436_ref014","volume-title":"The effective content of Ulm's theorem. Effective aspects of algebra","author":"Lin","year":"1981"},{"key":"S0022481200045436_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/BF01504333"},{"key":"S0022481200045436_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90013-7"},{"key":"S0022481200045436_ref008","volume-title":"Abelian groups","author":"Fuchs","year":"1958"},{"key":"S0022481200045436_ref016","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BFb0062858","volume-title":"Algebra and Logic, Lecture Notes in Mathematics","author":"Metakides","year":"1975"},{"key":"S0022481200045436_ref003","first-page":"363","volume":"34","author":"Dekker","year":"1969","journal-title":"Countable vector spaces with recursive operations. Part I"},{"key":"S0022481200045436_ref006","first-page":"117","volume-title":"Lecture Notes in Mathematics","author":"Feferman","year":"1975"},{"key":"S0022481200045436_ref007","first-page":"407","article-title":"Effective procedures in field theory","volume":"284","author":"Frohlich","year":"1955","journal-title":"Philosophical Transactions of the Royal Society of London, Section A"},{"key":"S0022481200045436_ref011","volume-title":"Infinite abelian groups","author":"Kaplansky","year":"1969"},{"key":"S0022481200045436_ref009","unstructured":"Harrington L. , Recursively presentable prime models, mimeograph, 1972."},{"key":"S0022481200045436_ref012","first-page":"165","article-title":"Zur Theorie der abelschen Gruppen von beliebiger M\u00e4chtigheit","volume":"9","author":"Kulikov","year":"1941","journal-title":"Matemati\u010deskit Sborni\u01d0"},{"key":"S0022481200045436_ref015","unstructured":"Lin C. , Recursively presented abelian groups: Effective p-group theory. II (to appear)."},{"key":"S0022481200045436_ref019","first-page":"307","volume":"44","author":"Millar","year":"1979","journal-title":"A complete decidable theory with two recursively presentable models"},{"key":"S0022481200045436_ref022","unstructured":"Remmel J. , Co-recursively enumerable structures, Ph.D. dissertation, Cornell University, 1974."},{"key":"S0022481200045436_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90015-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200045436","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,25]],"date-time":"2019-05-25T15:44:28Z","timestamp":1558799068000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200045436\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1981,9]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1981,9]]}},"alternative-id":["S0022481200045436"],"URL":"https:\/\/doi.org\/10.2307\/2273759","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1981,9]]}}}