Dominik Bernhardt, Tim Boykett, Alice Devillers, Johannes Flake, S. Glasby
Abstract We study the groups 𝐺 with the curious property that there exists an element k ∈ G kin G and a function f : G → G fcolon Gto G such that f ( x k ) = x f ( x ) f(xk)=xf(x) holds for all x ∈ G xin G . This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a 𝐽-group. Finite 𝐽-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a 𝐽-group if its nilpotency class 𝑐 satisfies c ⩽ 6 cleqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 p>2 and 𝐺 is a regular 𝑝-group or, more generally, a power-closed one (i.e., in each section and for each m ⩾ 1 mgeqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.
摘要研究了一类群𝐺，它们具有一个奇异的性质，即存在一个元素k∈G k in G和一个函数f: G→G f colon G to G，使得f(x)减去(x)减去f(x)减去f(xk)减去xf(x)对于所有x∈G x in G都成立。这一性质源于对群上的近环和输入输出自动机的研究。我们称具有此属性的组为𝐽-group。有限的𝐽-groups必须是奇阶的，因此是可解的。证明了奇数阶幂零群是一个𝐽-group，如果它的幂零类𝑐满足c≤6 c≤leqslant 6。如果𝐺是有限的𝑝-group，且p>2 p>2且p>2∑c-1 p^{2}>2c-1，则证明𝐺是𝐽-group。最后，如果p>2 p>2并且𝐺是一个规则的𝑝-group，或者更一般地说，是一个幂闭的𝑝-group(即，在每个部分中并且对于每个m小于1 m geqslant 1, p {m p^m} -幂的子集是一个子群)，那么我们证明𝐺是𝐽-group。
{"title":"The groups 𝐺 satisfying a functional equation 𝑓(𝑥𝑘) = 𝑥𝑓(𝑥) for some 𝑘 ∈ 𝐺","authors":"Dominik Bernhardt, Tim Boykett, Alice Devillers, Johannes Flake, S. Glasby","doi":"10.1515/jgth-2021-0158","DOIUrl":"https://doi.org/10.1515/jgth-2021-0158","url":null,"abstract":"Abstract We study the groups 𝐺 with the curious property that there exists an element k ∈ G kin G and a function f : G → G fcolon Gto G such that f ( x k ) = x f ( x ) f(xk)=xf(x) holds for all x ∈ G xin G . This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a 𝐽-group. Finite 𝐽-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a 𝐽-group if its nilpotency class 𝑐 satisfies c ⩽ 6 cleqslant 6 . If 𝐺 is a finite 𝑝-group, with p > 2 p>2 and p 2 > 2 c - 1 p^{2}>2c-1 , then we prove that 𝐺 is 𝐽-group. Finally, if p > 2 p>2 and 𝐺 is a regular 𝑝-group or, more generally, a power-closed one (i.e., in each section and for each m ⩾ 1 mgeqslant 1 , the subset of p m p^{m} -th powers is a subgroup), then we prove that 𝐺 is a 𝐽-group.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74740310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.
{"title":"On Whitehead’s cut vertex lemma","authors":"Rylee Alanza Lyman","doi":"10.1515/jgth-2022-0089","DOIUrl":"https://doi.org/10.1515/jgth-2022-0089","url":null,"abstract":"Abstract One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81128153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝜎 be a partition of the set of all primes, and let 𝔉 denote a hereditary formation. We describe all formations 𝔉 for which the 𝔉-hypercenter and the intersection of weak 𝐾-𝔉-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all 𝜎-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkov’s problem about the intersection of 𝔉-maximal subgroups and the 𝔉-hypercenter. As a corollary, we obtain Hall’s classical result about the hypercenter. We prove that the non-𝜎-nilpotent graph of a group is connected and its diameter is at most 3.
{"title":"On the 𝜎-nilpotent hypercenter of finite groups","authors":"V. I. Murashka, A. Vasil'ev","doi":"10.1515/jgth-2021-0138","DOIUrl":"https://doi.org/10.1515/jgth-2021-0138","url":null,"abstract":"Abstract Let 𝜎 be a partition of the set of all primes, and let 𝔉 denote a hereditary formation. We describe all formations 𝔉 for which the 𝔉-hypercenter and the intersection of weak 𝐾-𝔉-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all 𝜎-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkov’s problem about the intersection of 𝔉-maximal subgroups and the 𝔉-hypercenter. As a corollary, we obtain Hall’s classical result about the hypercenter. We prove that the non-𝜎-nilpotent graph of a group is connected and its diameter is at most 3.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79975569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 M 12 2M_{12} , M 11 M_{11} , and A 6 = O 2 ( M 10 ) A_{6}=O^{2}(M_{10}) over F 3 mathbb{F}_{3} , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway’s sporadic simple groups.
{"title":"Fusion systems realizing certain Todd modules","authors":"B. Oliver","doi":"10.1515/jgth-2022-0074","DOIUrl":"https://doi.org/10.1515/jgth-2022-0074","url":null,"abstract":"Abstract We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 M 12 2M_{12} , M 11 M_{11} , and A 6 = O 2 ( M 10 ) A_{6}=O^{2}(M_{10}) over F 3 mathbb{F}_{3} , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway’s sporadic simple groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84069989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.
{"title":"The number of locally invariant orderings of a group","authors":"I. Ba, A. Clay, I. Thompson","doi":"10.1515/jgth-2022-0126","DOIUrl":"https://doi.org/10.1515/jgth-2022-0126","url":null,"abstract":"Abstract We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80812269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝐺 be a finite group and p k p^{k} a prime power dividing | G | lvert Grvert . A subgroup 𝐻 of 𝐺 is said to be ℳ-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H K G=HK and H i K < G H_{i}K
摘要设𝐺为有限群，p k p^{k}为素数幂除以| G | lvert G rvert。如果𝐺存在一个子群𝐾，使得对于𝐻的每一个极大子群H i H＿i, G=H∑K G=HK且H i∑K
{"title":"On ℳ-supplemented subgroups","authors":"Yuedi Zeng","doi":"10.1515/jgth-2021-0195","DOIUrl":"https://doi.org/10.1515/jgth-2021-0195","url":null,"abstract":"Abstract Let 𝐺 be a finite group and p k p^{k} a prime power dividing | G | lvert Grvert . A subgroup 𝐻 of 𝐺 is said to be ℳ-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H K G=HK and H i K < G H_{i}K<G for every maximal subgroup H i H_{i} of 𝐻. In this paper, we complete the classification of the finite groups 𝐺 in which all subgroups of order p k p^{k} are ℳ-supplemented. In particular, we show that if k ≥ 2 kgeq 2 , then G / O p ′ ( G ) G/mathbf{O}_{p^{prime}}(G) is supersolvable with a normal Sylow 𝑝-subgroup and a cyclic 𝑝-complement.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84174199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we compute powers in the wreath product G ≀ S n Gwr S_{n} for any finite group 𝐺. For r ≥ 2 rgeq 2 a prime, consider ω r : G ≀ S n → G ≀ S n omega_{r}colon Gwr S_{n}to Gwr S_{n} defined by g ↦ g r gmapsto g^{r} . Let P r ( G ≀ S n ) := | ω r ( G ≀ S n ) | | G | n n ! P_{r}(Gwr S_{n}):=frac{lvertomega_{r}(Gwr S_{n})rvert}{lvert Grvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n Gwr S_{n} is an 𝑟-th power. We prove P r ( G ≀ S n + 1 ) = P r ( G ≀ S n ) P_{r}(Gwr S_{n+1})=P_{r}(Gwr S_{n}) for all n ≢ - 1 ( mod r ) nnotequiv-1 (mathrm{mod} r) if the order of 𝐺 is coprime to 𝑟. We also give a formula for the number of conjugacy classes that are 𝑟-th powers in G ≀ S n Gwr S_{n} .
抽象的这篇文章,我们《wreath鲍尔compute广告G≀S n G wr S_{}对于任何有限的𝐺集团。为r≥2 r geq a prime,认为ωr: G≀S n→G≀结肠G n omega_ {r的wr S_ {n}到G wr S_ (n):是由G↦G r G r mapsto G ^{}。让P r S(G≀n): = |ωS r(G≀n) | | G | nn !P_ {r} (G n wr S_ {}): = frac {lvert r omega_ {} (G wr S_ {n}) rvert} {lvert G rvert ^ {n, n !be a probability那randomly被选中元素》是G≀S n G wr S_{}是一个𝑟-th电源。我们证明P r S(G≀n + 1) = P r S(G≀n) P_ {r} (G wr S_ (n + 1)) = r P_ {} (G wr S_ {n})为所有n≢- 1(modr) n equiv-1音符(mathrm {mod} r)如果《𝐺是coprime到𝑟勋章。我们当家》也给a配方for conjugacy课堂这是鲍尔𝑟-th in G≀S n G wr S_{}。
{"title":"Powers in wreath products of finite groups","authors":"Rijubrata Kundu, Sudipa Mondal","doi":"10.1515/jgth-2021-0057","DOIUrl":"https://doi.org/10.1515/jgth-2021-0057","url":null,"abstract":"Abstract In this paper, we compute powers in the wreath product G ≀ S n Gwr S_{n} for any finite group 𝐺. For r ≥ 2 rgeq 2 a prime, consider ω r : G ≀ S n → G ≀ S n omega_{r}colon Gwr S_{n}to Gwr S_{n} defined by g ↦ g r gmapsto g^{r} . Let P r ( G ≀ S n ) := | ω r ( G ≀ S n ) | | G | n n ! P_{r}(Gwr S_{n}):=frac{lvertomega_{r}(Gwr S_{n})rvert}{lvert Grvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n Gwr S_{n} is an 𝑟-th power. We prove P r ( G ≀ S n + 1 ) = P r ( G ≀ S n ) P_{r}(Gwr S_{n+1})=P_{r}(Gwr S_{n}) for all n ≢ - 1 ( mod r ) nnotequiv-1 (mathrm{mod} r) if the order of 𝐺 is coprime to 𝑟. We also give a formula for the number of conjugacy classes that are 𝑟-th powers in G ≀ S n Gwr S_{n} .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76131877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=Dcup D^{-1} for some integer n ≥ 2 ngeq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ langle Krangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 ngeq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.
摘要设𝐾和𝐷是有限群𝐺的共轭类，并设K n=D∪D -1 K^{n}=D cup D^{-1}对于某整数n≥2 n geq 2。在这些假设下，我们推测⟨K⟩langle K rangle必须是𝐺的一个(正规的)可解的子群。最近研发。Camina已经证明了这个猜想对任何n≥4 n geq 4都是有效的，这是通过应用组合结果来完成的，其中主要涉及有限群中具有小倍的子集。在这篇笔记中，我们通过求助于其他的组合结果来解决n= 3n =3的情况，例如有限群中两个正态集积的基数的估计，以及一些最新的技术和定理。
{"title":"On powers of conjugacy classes in finite groups","authors":"A. Beltrán","doi":"10.1515/jgth-2021-0156","DOIUrl":"https://doi.org/10.1515/jgth-2021-0156","url":null,"abstract":"Abstract Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=Dcup D^{-1} for some integer n ≥ 2 ngeq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ langle Krangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 ngeq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90795215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.
{"title":"On the Tits alternative for cyclically presented groups with length-four positive relators","authors":"Shaun Isherwood, Gerald Williams","doi":"10.1515/jgth-2021-0131","DOIUrl":"https://doi.org/10.1515/jgth-2021-0131","url":null,"abstract":"Abstract We investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74186771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.
{"title":"The nilpotent genus of finitely generated residually nilpotent groups","authors":"N. O’Sullivan","doi":"10.1515/jgth-2022-0098","DOIUrl":"https://doi.org/10.1515/jgth-2022-0098","url":null,"abstract":"Abstract Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91257143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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