[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-268576-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-268576-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","on-clt-and-non-clt-groups","On CLT and non-CLT groups","","This note studies finite groups in the context of the Converse of Lagrange’s Theorem, distinguishing CLT groups from non-CLT groups. For every integer d≥2 that is not a prime power, it constructs a finite solvable group G such that d divides |G|, the prime divisors of |G| and d match, and G has no subgroup of order d. It also defines the CLT-degree, proves density results, and derives corollaries answering related questions.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/on-clt-and-non-clt-groups/268576/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/on-clt-and-non-clt-groups/268576.png","ImageObject",300,407,{"name":42,"@type":43},"Franzy","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",12,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is a CLT group and how is it related to the Converse of Lagrange’s Theorem?","Question",{"text":63,"@type":64},"A finite group is CLT if it has subgroups of every possible order dividing its size, meaning it satisfies the Converse of Lagrange’s Theorem. Otherwise it is non-CLT.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"For which integers d does the paper construct a solvable group G with no subgroup of order d?",{"text":68,"@type":64},"For every integer d≥2 that is not a prime power, the paper shows there exists a finite solvable group G such that d divides |G|, the prime divisors match, and G has no subgroup of order d.",{"name":70,"@type":61,"acceptedAnswer":71},"What is the CLT-degree dCLT(G) and what key property does it have?",{"text":72,"@type":64},"The CLT-degree dCLT(G) measures the proportion of divisors of |G| that occur as subgroup orders. It satisfies 0\u003CdCLT(G)≤1, is multiplicative for coprime-order products, and equals 1 exactly when G is CLT.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},268576,1789432528,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":97,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":112,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":118,"preview_page_count":4},2336478945519,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","arXiv :2403 .05774v1 [math .GR] 9 Mar 2024  \nOn CLT and non-CLT groups  \nMarius T􀀕arn􀀕auceanu March 9, 2024  \nAbstract  \nIn this note, we prove that for every integer d ≥ 2 which is not a prime power, there exists a 􀀌nite solvable group G such that d | |G| ,􀀙 (G) = 􀀙 (d) and G has no subgroup of order d. We also introduce the CLT-degree of a 􀀌nite group and answer two questions about it.  \nMSC2020 : Primary 20D60; Secondary 20D99, 20E99 .  \nKey words : CLT group, solvable group, Frobenius group.  \n1 Introduction  \nAll groups considered in this note are 􀀌nite. A group is said to be CLT if it possesses subgroups of every possible order (that is, it satis􀀌es the Converse of Lagrange’s Theorem) and non-CLT otherwise. It is well-known that CLT groups are solvable (see [11]) and that supersolvable groups are CLT (see [10]) . Moreover, the inclusion between the classes of CLT groups and solvable groups, as well as the inclusion between the classes of supersolvable groups and CLT groups are proper (see, for example, [3]) . Recall also the papers [1, 2, 13] which construct non-CLT groups of order p􀀋 q􀀌 with p, q primes and 􀀋,􀀌 ∈ N∗ .  \nThe 􀀌rst starting point for our work is given by the smallest examples of non-CLT groups, namely A4 and SL(2 , 3) . It is well-known that they have no subgroup of order 6 and 12, respectively. This leads to the following natural question:  \nGiven an integer d ≥ 2, is it possible to construct a group whose order is divisible by d, but having no subgroup of order d?  \nObviously, the answer to this question is ”NO” if d is a prime power. So, in what follows we will assume that d is a composite number and we will denote by 􀀙(d) the set of primes dividing d. Also, for a 􀀌nite group G we will denote 􀀙 (G) = 􀀙(|G|) .  \nOur 􀀌rst result shows that the answer to the above question is ”YES” for all composite numbers d.  \nTheorem 1.1 . For every integer d ≥ 2 which is not a prime power, there exists a 􀀌nite solvable group G such that d | |G| , 􀀙 (G) = 􀀙(d) and G has no subgroup of order d.  \nThe second starting point for our work is given by a question of Martino Garonzi on MathOver􀀍ow [4]:  \nIs there a constant c > 0 such that D􀀜(~~ ~~|(GG|)) > c for any 􀀌nite group G? Here D (G) denotes the number of divisors d of |G| for which there exists a subgroup of G of order d and 􀀜 (|G|) denotes the number of all divisors of  \n|G| . Note that this question remained unanswered for more than ten years. It suggests to consider the function  \nD (G) dCLT (G) = ,  \n􀀜 (|G|)  \nwhich will be called the CLT-degree of G. Clearly, dCLT (G) measures the probability of a 􀀌nite group G to be CLT. It is easy to see that this new function satis􀀌es the following properties:  \n- 0 \u003C dCLT (G) ≤ 1, for any 􀀌nite group G. Moreover, dCLT (G) = 1 if and only if G is CLT.  \n- dCLT is multiplicative, that is dCLT (G1 ×G2 ) = dCLT (G1 )dCLT (G2 ), for any 􀀌nite groups G 1 , G2 of coprime orders.  \n- If N is a normal subgroup of G, then  \n 􀀜 (|G|)  dCLT (G/N) ≤ dCLT (G) .  \n􀀜 (|G|/|N|)  \n- If |G| = pn11 · · ·pnkk with k ≥ 2, then  \ndCLT (G) ≥ PQki~~1~~(nnii~~ ~~21) .  \nMoreover, we have equality if and only if G has all proper subgroups of prime power order. Such a group is either cyclic of order pq, where p and q are distinct primes, or a semidirect product Cnp ⋊ Cq , where Cq acts irreducibly on Cnp . Thus  \nn + 3  \ndCLT (Cnp ⋊ Cq ) =  \n2n + 2  \nand, in particular, dCLT (A4 ) = ~~5~~6 .  \nLet G be the class of all 􀀌nite groups. Our second result is stated as follows.  \nTheorem 1 .2. The set  \nIm(dCLT ) = {dCLT (G) | G ∈ G}  \nis dense in [0 , 1] .  \nObviously, Theorem 1 .2 gives a negative answer to Garonzi’s question. Corollary 1.3 . There is no constant c > 0 such that D􀀜(~~ ~~|(GG|)) > c for any 􀀌nite group G.  \nWe also infer that:  \nCorollary 1.4 . There is no constant c \u003C 1 such that dCLT (G) ≤ c for any 􀀌nite non-CLT group G.  \nNote that an example of a family of 􀀌nite non-CLT groups (Gn )n≥1 with limn→∞ dCLT (Gn ) = 1 will be given at the end","cbCaio2eGvsZAsZe","https://ap.wps.com/l/cbCaio2eGvsZAsZe","pdf",107609,"English","# Introduction\n## Proof of Theorem 1.1\n### Auxiliary results\n## Proof of Theorem 1.1 (structure)","[{\"question\":\"What is a CLT group and how is it related to the Converse of Lagrange’s Theorem?\",\"answer\":\"A finite group is CLT if it has subgroups of every possible order dividing its size, meaning it satisfies the Converse of Lagrange’s Theorem. Otherwise it is non-CLT.\"},{\"question\":\"For which integers d does the paper construct a solvable group G with no subgroup of order d?\",\"answer\":\"For every integer d≥2 that is not a prime power, the paper shows there exists a finite solvable group G such that d divides |G|, the prime divisors match, and G has no subgroup of order d.\"},{\"question\":\"What is the CLT-degree dCLT(G) and what key property does it have?\",\"answer\":\"The CLT-degree dCLT(G) measures the proportion of divisors of |G| that occur as subgroup orders. It satisfies 0\\u003cdCLT(G)≤1, is multiplicative for coprime-order products, and equals 1 exactly when G is CLT.\"}]","On CLT and non-CLT groups | PDF"]