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Finite Groups with a Soluble Group of Coprime Automorphisms Whose Fixed Points Have Bounded Engel Sinks 定点具有有界恩格尔下沉的可溶同素自动形群有限群
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-01-04 DOI: 10.1007/s10469-023-09727-w
E. I. Khukhro, P. Shumyatsky
{"title":"Finite Groups with a Soluble Group of Coprime Automorphisms Whose Fixed Points Have Bounded Engel Sinks","authors":"E. I. Khukhro,&nbsp;P. Shumyatsky","doi":"10.1007/s10469-023-09727-w","DOIUrl":"10.1007/s10469-023-09727-w","url":null,"abstract":"<p>Suppose that a finite group <i>G</i> admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer <i>m</i>, every element of the centralizer <i>C</i><sub><i>G</i></sub>(<i>A</i>) has a left Engel sink of cardinality at most <i>m</i> (or a right Engel sink of cardinality at most <i>m</i>), then <i>G</i> has a subgroup of (|<i>A</i>|,<i>m</i>)-bounded index which has Fitting height at most 2α(<i>A</i>) + 2, where α(<i>A</i>) is the composition length of <i>A</i>. We also prove that if, for some positive integer <i>r</i>, every element of the centralizer <i>C</i><sub><i>G</i></sub>(<i>A</i>) has a left Engel sink of rank at most <i>r</i> (or a right Engel sink of rank at most <i>r</i>), then <i>G</i> has a subgroup of (|<i>A</i>|, <i>r</i>)-bounded index which has Fitting height at most 4α(A) + 4α(A) + 3. Here, a left Engel sink of an element g of a group <i>G</i> is a set <i>𝔈</i> (<i>g</i>) such that for every <i>x</i> ∈ <i>G</i> all sufficiently long commutators [...[[<i>x</i>, <i>g</i>], <i>g</i>], . . . , <i>g</i>] belong to <i>𝔈</i> (<i>g</i>). (Thus, g is a left Engel element precisely when we can choose (g) = {1}.) A right Engel sink of an element g of a group <i>G</i> is a set <i>ℜ</i>(<i>g</i>) such that for every <i>x</i> ∈ <i>G</i> all sufficiently long commutators [...[[g, x], x], . . . , x] belong to <i>ℜ</i>(<i>g</i>). Thus, <i>g</i> is a right Engel element precisely when we can choose <i>ℜ</i>(<i>g</i>) = {1}.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"80 - 93"},"PeriodicalIF":0.4,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139093209","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}
引用次数: 0
Primitive Prime Divisors of Orders of Suzuki–Ree Groups 铃木李群阶的原始素除数
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-01-03 DOI: 10.1007/s10469-023-09722-1
M. A. Grechkoseeva
{"title":"Primitive Prime Divisors of Orders of Suzuki–Ree Groups","authors":"M. A. Grechkoseeva","doi":"10.1007/s10469-023-09722-1","DOIUrl":"10.1007/s10469-023-09722-1","url":null,"abstract":"<p>There is a well-known factorization of the number 2<sup>2<i>m</i></sup> + 1<i>,</i> with m odd, related to the orders of tori of simple Suzuki groups: 2<sup>2<i>m</i></sup> +1 is a product of <i>a</i> = 2<sup><i>m</i></sup> + 2<sup>(<i>m</i>+1)<i>/</i>2</sup> +1 and <i>b</i> = 2<sup><i>m</i></sup><i> −</i> 2<sup>(<i>m</i>+1)<i>/</i>2</sup> + 1. By the Bang–Zsigmondy theorem, there is a primitive prime divisor of 2<sup>4<i>m</i></sup><i> −</i> 1, that is, a prime r that divides 2<sup>4<i>m</i></sup> − 1 and does not divide 2<sup><i>i</i></sup><i> −</i> 1 for any 1 ≤ <i>i &lt;</i> 4<i>m.</i> It is easy to see that r divides 2<sup>2<i>m</i></sup> + 1, and so it divides one of the numbers <i>a</i> and <i>b.</i> It is proved that for every <i>m &gt;</i> 5, each of <i>a, b</i> is divisible by some primitive prime divisor of 2<sup>4<i>m</i></sup><i> −</i> 1. Similar results are obtained for primitive prime divisors related to the simple Ree groups. As an application, we find the independence and 2-independence numbers of the prime graphs of almost simple Suzuki–Ree groups.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"41 - 49"},"PeriodicalIF":0.4,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139096709","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}
引用次数: 0
Generic Types and Generic Elements in Divisible Rigid Groups 可分刚性群中的通用类型和通用元素
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-01-03 DOI: 10.1007/s10469-023-09726-x
A. G. Myasnikov, N. S. Romanovskii
{"title":"Generic Types and Generic Elements in Divisible Rigid Groups","authors":"A. G. Myasnikov,&nbsp;N. S. Romanovskii","doi":"10.1007/s10469-023-09726-x","DOIUrl":"10.1007/s10469-023-09726-x","url":null,"abstract":"<p>A group <i>G</i> is said to be m-rigid if it contains a normal series of the form <i>G</i> = <i>G</i><sub>1</sub> &gt; <i>G</i><sub>2</sub> &gt; . . . &gt; <i>G</i><sub><i>m</i></sub> &gt; <i>G</i><sub><i>m+</i>1</sub> = 1, whose quotients <i>G</i><sub><i>i</i></sub>/<i>G</i><sub><i>i+</i>1</sub> are Abelian and, treated as (right) ℤ[<i>G</i>/<i>G</i><sub><i>i</i></sub>]-modules, are torsion-free. A rigid group <i>G</i> is said to be divisible if elements of the quotient <i>ρ</i><sub><i>i</i></sub>(<i>G</i>)/<i>ρ</i><sub><i>i+</i>1</sub>(<i>G</i>) are divisible by nonzero elements of the ring ℤ[<i>G</i>/<i>ρ</i><sub><i>i</i></sub>(<i>G</i>)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group <i>G</i>.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"72 - 79"},"PeriodicalIF":0.4,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139096549","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}
引用次数: 0
Finite 4-Primary Groups with Disconnected Gruenberg–Kegel Graph Containing a Triangle 包含三角形的断开格伦伯格-凯格尔图的有限四元组
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-01-03 DOI: 10.1007/s10469-023-09724-z
A. S. Kondrat’ev
{"title":"Finite 4-Primary Groups with Disconnected Gruenberg–Kegel Graph Containing a Triangle","authors":"A. S. Kondrat’ev","doi":"10.1007/s10469-023-09724-z","DOIUrl":"10.1007/s10469-023-09724-z","url":null,"abstract":"<p>We give a description of finite 4-primary groups with disconnected Gruenberg–Kegel graph containing a triangle. As a corollary, finite groups whose Gruenberg–Kegel graph coincides with the Gruenberg–Kegel graph of <sup>3</sup>D<sub>4</sub>(2) are exemplified, which generalizes V. D. Mazurov’ description of finite groups isospectral to the group <sup>3</sup>D<sub>4</sub>(2).</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"54 - 65"},"PeriodicalIF":0.4,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094141","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}
引用次数: 0
On p-Index Extremal Groups 关于 p 指数极值群
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-01-03 DOI: 10.1007/s10469-023-09728-9
A. V. Vasil’ev, I. B. Gorshkov
{"title":"On p-Index Extremal Groups","authors":"A. V. Vasil’ev,&nbsp;I. B. Gorshkov","doi":"10.1007/s10469-023-09728-9","DOIUrl":"10.1007/s10469-023-09728-9","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"94 - 99"},"PeriodicalIF":0.4,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409583","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}
引用次数: 0
Shunkov Groups Saturated with Almost Simple Groups 几乎简单群饱和的 Shunkov 群
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2023-12-28 DOI: 10.1007/s10469-023-09725-y
N. V. Maslova, A. A. Shlepkin
{"title":"Shunkov Groups Saturated with Almost Simple Groups","authors":"N. V. Maslova,&nbsp;A. A. Shlepkin","doi":"10.1007/s10469-023-09725-y","DOIUrl":"10.1007/s10469-023-09725-y","url":null,"abstract":"<p>A group <i>G</i> is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups <i>H</i> in the factor group <i>N</i><sub><i>G</i></sub>(<i>H</i>)/<i>H</i>, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group <i>G</i> which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution <i>z</i> with the property that the centralizer <i>C</i><sub><i>G</i></sub>(<i>z</i>) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group <i>G</i> that is saturated with finite almost simple groups and contains an involution <i>z</i> with the property that the centralizer <i>C</i><sub><i>G</i></sub>(<i>z</i>) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"66 - 71"},"PeriodicalIF":0.4,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139064294","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}
引用次数: 0
Unsolvability of Finite Groups Isospectral to the Automorphism Group of the Second Sporadic Janko Group 与第二时空扬科群的自变群同谱的有限群的不可解性
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2023-12-28 DOI: 10.1007/s10469-023-09723-0
A. Kh. Zhurtov, D. V. Lytkina, V. D. Mazurov
{"title":"Unsolvability of Finite Groups Isospectral to the Automorphism Group of the Second Sporadic Janko Group","authors":"A. Kh. Zhurtov,&nbsp;D. V. Lytkina,&nbsp;V. D. Mazurov","doi":"10.1007/s10469-023-09723-0","DOIUrl":"10.1007/s10469-023-09723-0","url":null,"abstract":"<p>For a finite group G, the spectrum is the set ω(G) of element orders of the group G. The spectrum of G is closed under divisibility and is therefore uniquely determined by the set μ(G) consisting of elements of ω(G) that are maximal with respect to divisibility. We prove that a finite group isospectral to Aut(J<sub>2</sub>) is unsolvable.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"50 - 53"},"PeriodicalIF":0.4,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139064341","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}
引用次数: 0
Sessions of the Seminar “Algebra i Logika” 代数与逻辑 "研讨会课程
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2023-12-16 DOI: 10.1007/s10469-023-09729-8
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-023-09729-8","DOIUrl":"10.1007/s10469-023-09729-8","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 1","pages":"100 - 101"},"PeriodicalIF":0.4,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412030","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}
引用次数: 0
Families of Permutations and Ideals of Turing Degrees 排列族与图灵度理想
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2023-12-04 DOI: 10.1007/s10469-023-09714-1
A. S. Morozov, V. G. Puzarenko, M. Kh. Faizrachmanov
{"title":"Families of Permutations and Ideals of Turing Degrees","authors":"A. S. Morozov,&nbsp;V. G. Puzarenko,&nbsp;M. Kh. Faizrachmanov","doi":"10.1007/s10469-023-09714-1","DOIUrl":"10.1007/s10469-023-09714-1","url":null,"abstract":"<p>Families 𝒫<sub>I</sub> consisting of permutations of the natural numbers ω whose degrees belong to an ideal I of Turing degrees, as well as their jumps <span>({mathcal{P}}_{mathrm{I}}{prime})</span>, are studied. For any countable Turing ideal I, the degree spectra of families 𝒫<sub>I</sub> and their jumps <span>({mathcal{P}}_{mathrm{I}}{prime})</span> are described. For some ideals I generated by c.e. degrees, the spectra of families 𝒫<sub>I</sub> are defined.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"481 - 490"},"PeriodicalIF":0.4,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529801","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}
引用次数: 0
Normal Companions of Intuitionistic Modal Logics 直觉模态逻辑的正规伴侣
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2023-11-23 DOI: 10.1007/s10469-023-09712-3
S. A. Drobyshevich
{"title":"Normal Companions of Intuitionistic Modal Logics","authors":"S. A. Drobyshevich","doi":"10.1007/s10469-023-09712-3","DOIUrl":"10.1007/s10469-023-09712-3","url":null,"abstract":"<p>Previously, Došen and Božić introduced four independent intuitionistic modal logics, one for each of four types of modal operators—necessity <i>N</i>, possibility <i>P</i>, impossibility <i>Im</i>, and unnecessity <i>Un</i>. These logics are denoted <i>HKM</i>, where <i>M</i> ∈ {<i>N</i>, <i>P</i>, <i>Un</i>, <i>Im</i>}. Interest in treating the four types of modal operators separately is associated with just the fact that these cannot be reduced to each other over intuitionistic logic. Here we study extensions of logics <i>HKM</i> that have normal companions. It turns out that all extensions of the logics <i>HKN</i> and <i>HKUn</i> possess normal companions. For the extensions of <i>HKP</i> and <i>HKIm</i>, we obtain a criterion for the existence of normal companions, which is postulated as the presence of some modal law of double negation. Also we show how adding of this law influences expressive capacities of a logic. Of particular interest is the result saying that extensions of <i>HKP</i> and <i>HKIm</i> have normal companions only if they are definitionally equivalent to those of <i>HKN</i> and <i>HKUn</i> respectively. This result is one more example of the differences in behavior of the four types of modal operators over intuitionistic logic.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"445 - 465"},"PeriodicalIF":0.4,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529756","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}
引用次数: 0
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