{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T00:53:57Z","timestamp":1648688037593},"reference-count":14,"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":10784,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,9]]},"abstract":"<jats:p>A countable group is \u2135<jats:sub>0<\/jats:sub>-<jats:italic>categorical<\/jats:italic> if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable \u2135<jats:sub>0<\/jats:sub>-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).<\/jats:p><jats:p>Let <jats:italic>K<\/jats:italic> be a finite nonabelian simple group. In \u00a73 we construct from <jats:italic>K<\/jats:italic> two denumberable groups, <jats:italic>K<\/jats:italic>* and <jats:italic>K<\/jats:italic><jats:sup>#<\/jats:sup>, and use them to classify countable direct limits of finite Cartesian powers of <jats:italic>K<\/jats:italic> (Proposition 1). The results of \u00a73 are used in \u00a74 to show that K* and K<jats:sup>#<\/jats:sup> are \u2135<jats:sub>0<\/jats:sub>-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, <jats:italic>K<\/jats:italic>* and <jats:italic>K<\/jats:italic><jats:sup>#<\/jats:sup> are counterexamples to the conjecture of U. Feigner [3, p. 309] that \u2135<jats:sub>0<\/jats:sub>-categorical groups are FC-solvable. Also in \u00a74 the results mentioned in the first paragraph are proved. \u00a72 is devoted to preliminary lemmas about finite Cartesian powers of <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>A <jats:italic>characteristic subgroup<\/jats:italic> of a group <jats:italic>G<\/jats:italic> is a subgroup which is mapped onto itself by all automorphisms of <jats:italic>G<\/jats:italic>. <jats:italic>G<\/jats:italic> is <jats:italic>characteristically simple<\/jats:italic> if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If <jats:italic>G<\/jats:italic> is \u2135<jats:sub>0<\/jats:sub>-categorical, then as the automorphism group of <jats:italic>G<\/jats:italic> has only a finite number of orbits, <jats:italic>G<\/jats:italic> has only a finite number of characteristic subgroups.<\/jats:p>","DOI":"10.2307\/2274143","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:09:38Z","timestamp":1146953378000},"page":"900-907","source":"Crossref","is-referenced-by-count":0,"title":["Characteristically simple \u2135<sub>0<\/sub>-categorical groups"],"prefix":"10.1017","volume":"49","author":[{"given":"Robert H.","family":"Gilman","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043048_ref012","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511661884.026"},{"key":"S0022481200043048_ref010","volume-title":"Group theory","author":"Scott","year":"1964"},{"key":"S0022481200043048_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(79)90230-8"},{"key":"S0022481200043048_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70851-6"},{"key":"S0022481200043048_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(73)90092-6"},{"key":"S0022481200043048_ref011","first-page":"195","article-title":"On \u03c90-categoricity of powers","volume":"17","author":"Waszkiewicz","year":"1969","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200043048_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF02485820"},{"key":"S0022481200043048_ref002","volume-title":"Model theory","author":"Chang","year":"1977"},{"key":"S0022481200043048_ref003","first-page":"301","volume-title":"Logic Colloquium '76","author":"Felgner","year":"1977"},{"key":"S0022481200043048_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-64981-3"},{"key":"S0022481200043048_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71100-5"},{"key":"S0022481200043048_ref009","first-page":"591","article-title":"Model companions for \u21350-categorical theories","volume":"39","author":"Saracino","year":"1973","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200043048_ref013","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100059442"},{"key":"S0022481200043048_ref014","unstructured":"Apps A. B. , On the structure of \u21350-categorical groups (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043048","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T19:49:17Z","timestamp":1558640957000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043048\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,9]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1984,9]]}},"alternative-id":["S0022481200043048"],"URL":"https:\/\/doi.org\/10.2307\/2274143","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,9]]}}}