{"title":"CONGRUENCES FOR RANKS OF PARTITIONS","authors":"RENRONG MAO","doi":"10.1017/s0004972723001454","DOIUrl":"https://doi.org/10.1017/s0004972723001454","url":null,"abstract":"Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001454_inline1.png\" /> <jats:tex-math> $5$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for ranks of partitions.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS","authors":"NADIA TAGHIPOUR, SHAMILA BAYATI, FARHAD RAHMATI","doi":"10.1017/s0004972723001363","DOIUrl":"https://doi.org/10.1017/s0004972723001363","url":null,"abstract":"It is well known that the edge ideal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline1.png\" /> <jats:tex-math> $I(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of a simple graph <jats:italic>G</jats:italic> has linear quotients if and only if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline2.png\" /> <jats:tex-math> $G^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline3.png\" /> <jats:tex-math> $G^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline4.png\" /> <jats:tex-math> $I(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has homological linear quotients results in a graph with the same property. In particular, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline5.png\" /> <jats:tex-math> $I(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has homological linear quotients when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline6.png\" /> <jats:tex-math> $G^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline7.png\" /> <jats:tex-math> $I(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has homological linear quotients for every graph <jats:italic>G</jats:italic> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline8.png\" /> <jats:tex-math> $G^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001363_inline9.png\" /> <jats:tex-math> $lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-min","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"MAXIMAL SUBSEMIGROUPS OF INFINITE SYMMETRIC GROUPS","authors":"SUZANA MENDES-GONÇALVES, R. P. SULLIVAN","doi":"10.1017/s0004972723001375","DOIUrl":"https://doi.org/10.1017/s0004972723001375","url":null,"abstract":"Brazil <jats:italic>et al</jats:italic>. [‘Maximal subgroups of infinite symmetric groups’, <jats:italic>Proc. Lond. Math. Soc. (3)</jats:italic>68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001375_inline1.png\" /> <jats:tex-math> $G(X)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> defined on an infinite set <jats:italic>X</jats:italic>. It is easy to see that, in this case, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001375_inline2.png\" /> <jats:tex-math> $G(X)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001375_inline3.png\" /> <jats:tex-math> $G(X)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We provide infinitely many examples of such semigroups.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, YONG YANG
{"title":"ON THE CHARACTERISATION OF ALTERNATING GROUPS BY CODEGREES","authors":"MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, YONG YANG","doi":"10.1017/s0004972723001429","DOIUrl":"https://doi.org/10.1017/s0004972723001429","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a finite group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline1.png\" /> <jats:tex-math> $mathrm {Irr}(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> the set of all irreducible complex characters of <jats:italic>G</jats:italic>. Define the codegree of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline2.png\" /> <jats:tex-math> $chi in mathrm {Irr}(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline3.png\" /> <jats:tex-math> $mathrm {cod}(chi ):={|G:mathrm {ker}(chi ) |}/{chi (1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline4.png\" /> <jats:tex-math> $mathrm {cod}(G):={mathrm {cod}(chi ) mid chi in mathrm {Irr}(G)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the codegree set of <jats:italic>G</jats:italic>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline5.png\" /> <jats:tex-math> $mathrm {A}_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an alternating group of degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline6.png\" /> <jats:tex-math> $n ge 5$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline7.png\" /> <jats:tex-math> $mathrm {A}_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is determined up to isomorphism by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001429_inline8.png\" /> <jats:tex-math> $operatorname {cod}(mathrm {A}_n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"COMPLETE EMBEDDINGS OF GROUPS","authors":"MARTIN R. BRIDSON, HAMISH SHORT","doi":"10.1017/s0004972723001442","DOIUrl":"https://doi.org/10.1017/s0004972723001442","url":null,"abstract":"Every countable group <jats:italic>G</jats:italic> can be embedded in a finitely generated group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline1.png\" /> <jats:tex-math> $G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that is hopfian and <jats:italic>complete</jats:italic>, that is, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline2.png\" /> <jats:tex-math> $G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has trivial centre and every epimorphism <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline3.png\" /> <jats:tex-math> $G^*to G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an inner automorphism. Every finite subgroup of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline4.png\" /> <jats:tex-math> $G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is conjugate to a finite subgroup of <jats:italic>G</jats:italic>. If <jats:italic>G</jats:italic> has a finite presentation (respectively, a finite classifying space), then so does <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline5.png\" /> <jats:tex-math> $G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Our construction of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001442_inline6.png\" /> <jats:tex-math> $G^*$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"BOUNDING ZETA ON THE 1-LINE UNDER THE PARTIAL RIEMANN HYPOTHESIS","authors":"ANDRÉS CHIRRE","doi":"10.1017/s0004972723001338","DOIUrl":"https://doi.org/10.1017/s0004972723001338","url":null,"abstract":"<p>We provide explicit bounds for the Riemann zeta-function on the line <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109124905552-0701:S0004972723001338:S0004972723001338_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {Re},{s}=1$</span></span></img></span></span>, assuming that the Riemann hypothesis holds up to height <span>T</span>. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139414794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
KEVIN BEANLAND, DMITRIY GOROVOY, JĘDRZEJ HODOR, DANIIL HOMZA
{"title":"COUNTING UNIONS OF SCHREIER SETS","authors":"KEVIN BEANLAND, DMITRIY GOROVOY, JĘDRZEJ HODOR, DANIIL HOMZA","doi":"10.1017/s0004972723001326","DOIUrl":"https://doi.org/10.1017/s0004972723001326","url":null,"abstract":"A subset of positive integers <jats:italic>F</jats:italic> is a Schreier set if it is nonempty and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline1.png\" /> <jats:tex-math> $|F|leqslant min F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (here <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline2.png\" /> <jats:tex-math> $|F|$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the cardinality of <jats:italic>F</jats:italic>). For each positive integer <jats:italic>k</jats:italic>, we define <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline3.png\" /> <jats:tex-math> $kmathcal {S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as the collection of all the unions of at most <jats:italic>k</jats:italic> Schreier sets. Also, for each positive integer <jats:italic>n</jats:italic>, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline4.png\" /> <jats:tex-math> $(kmathcal {S})^n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the collection of all sets in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline5.png\" /> <jats:tex-math> $kmathcal {S}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with maximum element equal to <jats:italic>n</jats:italic>. It is well known that the sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline6.png\" /> <jats:tex-math> $(|(1mathcal {S})^n|)_{n=1}^infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001326_inline7.png\" /> <jats:tex-math> $(|(kmathcal {S})^n|)_{n=1}^infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies a linear recurrence for every positive <jats:italic>k</jats:italic>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS","authors":"LIJUAN HE, HENG LV, GUIYUN CHEN","doi":"10.1017/s0004972723001168","DOIUrl":"https://doi.org/10.1017/s0004972723001168","url":null,"abstract":"Suppose that <jats:italic>G</jats:italic> is a finite solvable group. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline1.png\" /> <jats:tex-math> $t=n_c(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the number of orders of nonnormal subgroups of <jats:italic>G</jats:italic>. We bound the derived length <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline2.png\" /> <jats:tex-math> $dl(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in terms of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline3.png\" /> <jats:tex-math> $n_c(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. If <jats:italic>G</jats:italic> is a finite <jats:italic>p</jats:italic>-group, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline4.png\" /> <jats:tex-math> $|G'|leq p^{2t+1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline5.png\" /> <jats:tex-math> $dl(G)leq lceil log _2(2t+3)rceil $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. If <jats:italic>G</jats:italic> is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline6.png\" /> <jats:tex-math> $|G'|$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is less than <jats:italic>t</jats:italic> and that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001168_inline7.png\" /> <jats:tex-math> $dl(G)leq lfloor 2(t+1)/3rfloor +1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"INTERSECTING THE TORSION OF ELLIPTIC CURVES","authors":"NATALIA GARCIA-FRITZ, HECTOR PASTEN","doi":"10.1017/s000497272300134x","DOIUrl":"https://doi.org/10.1017/s000497272300134x","url":null,"abstract":"Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, <jats:italic>Diophantine Geometry</jats:italic>, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline1.png\" /> <jats:tex-math> $E_1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline2.png\" /> <jats:tex-math> $E_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> along with even degree-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline3.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> maps <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline4.png\" /> <jats:tex-math> $pi _jcolon E_jto mathbb {P}^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having different branch loci, the intersection of the image of the torsion points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline5.png\" /> <jats:tex-math> $E_1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline6.png\" /> <jats:tex-math> $E_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> under their respective <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline7.png\" /> <jats:tex-math> $pi _j$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272300134X_inline8.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna th","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}