[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82980-en":3,"doc-seo-82980-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82980,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Game Conductors of Finite Groups: Determinantal Torsion from Structured Payoff Probes","A finite group G together with a structured payoff probe ϕ defines an integer payoff-difference lattice Mϕ(G) and a conductor Cϕ(G), recording the primes where the lattice drops rank modulo p. For any nonabelian CA-group, the commuting conductor satisfies Ccomm(G)=rad(b−1), with b the number of maximal abelian subgroups, so conductor primes can escape the prime support of |G|. Commuting and character probes yield complementary prime behavior, with exhaustive computations and Smith-torsion analysis guiding a general classification program for centralizer-type incidence matrices.","arXiv :2607 .05698v2 [math .GR] 12 Jul 2026  \nGAME CONDUCTORS OF FINITE GROUPS: DETERMINANTAL TORSION FROM STRUCTURED PAYOFF PROBES  \nMATTHEW FRIED  \nAbstract. We attach to a finite group G and a structured payoff probe ϕ an integer payoffdifference lattice Mϕ (G) and its conductor Cϕ (G): the primes at which Mϕ (G) loses rank modulo  \np. Our main result is an exact computation: for any nonabelian CA-group the commuting conductor is rad (b − 1), where b is the number of maximal abelian subgroups. In particular, commuting-conductor primes need not divide |G|: the prime 3 occurs for a 2-group of order 64 with b = 7 . The commuting Smith spectrum is an invariant of the isoclinism class and obeysan exact direct-product law, giving Ccomm (G × H) = Ccomm (G) ∪ Ccomm (H) unconditionally.  \nA Galois-orbit-trace character probe reads a complementary layer: an index-2 subgroup forces  \n2 ∈ Cchar (G) while the odd-prime analogue fails for every odd prime, and Ccomm (D2q) = {q} , Cchar (D2q) = {2} for all odd primes q. Unconditionally Cchar (G) ⊆ rad (|G|), so the escape from the group order is exclusive to the commuting probe. Exhaustive exact computation (|G| ≤ 128 commuting, |G| ≤ 64 character) and a deformation-family analysis support the general program:  \nclassify the Smith torsion of the compressed centralizer-type incidence matrix BG .  \n1. Introduction  \nIn a 64 × 64 zero–one matrix record which pairs of elements of a group of order 64 commute, subtract rows, and ask modulo which primes the resulting integer lattice loses rank. For 95 .5% of the 3 ,349 nonabelian groups of order at most 128, computed exhaustively below, the answer is the set of primes dividing |G′|, the order of the derived subgroup. But for SmallGroup(64, 73) the answer is {2, 3}: a 2-group whose commuting structure detects the prime 3, a prime dividing neither the group order nor any character degree. The reason, proved below, is that the invariant computes not an order but a count: this group has b = 7 maximal abelian subgroups, and the lattice’s torsion is exactly b − 1 = 6 .  \nThis paper studies that invariant in general. The motivating question is classical: how much of a finite group is determined by its weakest relational data, the commuting relation, and by its coarsest character data, the rational character table? Our answer is a new place to look: not ranks or spectra over Q, but the integer Smith torsion of the associated difference lattices, read prime by prime. A game on a finite group G assigns a payoff to each ordered pair drawn from index sets attached to G (elements, conjugacy classes, or irreducible characters); the differences of payoff rows span an integer lattice Mϕ(G), and its conductor Cϕ(G) is the set of primes at which the lattice loses rank modulo p. The linear-algebraic engine is classical, Smith normal forms of incidence-type matrices [25, 24], and the conductor is a bad-reduction locus in the spirit of reduction-modulo-p techniques elsewhere in algebra [11]; what is new is the source of the lattice, not the technique. Our contribution is the correspondence and its laws: a game on a group carries a conductor; the conductor is tunable, with different games reading different structural layers; for the commuting game it is governed by a small centralizer-type incidence  \nDate: July 14, 2026 .  \n2020 Mathematics Subject Classification. 20C15, 20D60, 15A21, 05E16, 11C20 .  \nKey words and phrases. Smith normal form, determinantal divisor, commuting graph, CA-group, maximal abelian subgroup, character table, rank-drop conductor, experimental group theory.  \n\n| Proved | Theorems 2.2 , 3.1 , 4.3 , 4.6 , 7.1 , 7.3 , 9.1; Corollaries 3.2 , 4.8 , 9.4; existence in |\n| --- | --- |\n|  | Corollary 3.4; the reduction in Proposition 4.2; Proposition 9.2 (classical) |\n| Computed | Propositions 5.1 , 5.2 , 8.1 , 8.3 , 8.4; the tabulated spectra in Proposition 4.2; minimality of the order in Corollary 3.4; Tables 2 and 3 |\n| Conjectural | Conjecture 6.1 ","cbCaivs8CjWWMUbY","https://ap.wps.com/l/cbCaivs8CjWWMUbY","pdf",432248,3,1,17,"English","en",105,"# Introduction\n## Game and payoff probes\n## Conductor from rank drop\n## Main structural theorems\n## Classification program and computations","[{\"question\":\"What are payoff-difference lattices and conductors in this framework?\",\"answer\":\"Given a finite group G and a structured payoff probe ϕ, the payoff differences generate an integer lattice Mϕ(G). The conductor Cϕ(G) is the set of primes p where this lattice loses rank modulo p.\"},{\"question\":\"What is the main commuting-conductor formula for nonabelian CA-groups?\",\"answer\":\"For any nonabelian CA-group, the commuting conductor is Ccomm(G)=rad(b−1), where b is the number of maximal abelian subgroups.\"},{\"question\":\"Why can the conductor include primes that do not divide |G|?\",\"answer\":\"In the commuting probe, torsion is controlled by the count b−1 of maximal abelian subgroups rather than directly by |G| or character degrees. For example, prime 3 appears in a group of order 64 where 3 does not divide |G|.\"}]",1784184440,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"game-conductors-of-finite-groups-determinantal-torsion-from-structured-payoff-probes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/game-conductors-of-finite-groups-determinantal-torsion-from-structured-payoff-probes/82980/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are payoff-difference lattices and conductors in this framework?","Question",{"text":75,"@type":76},"Given a finite group G and a structured payoff probe ϕ, the payoff differences generate an integer lattice Mϕ(G). The conductor Cϕ(G) is the set of primes p where this lattice loses rank modulo p.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main commuting-conductor formula for nonabelian CA-groups?",{"text":80,"@type":76},"For any nonabelian CA-group, the commuting conductor is Ccomm(G)=rad(b−1), where b is the number of maximal abelian subgroups.",{"name":82,"@type":73,"acceptedAnswer":83},"Why can the conductor include primes that do not divide |G|?",{"text":84,"@type":76},"In the commuting probe, torsion is controlled by the count b−1 of maximal abelian subgroups rather than directly by |G| or character degrees. 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