Pub Date : 2019-02-01DOI: 10.22108/IJGT.2019.115131.1526
L. A. Kurdachenko, P. Longobardi, M. Maj
We study an inverse problem of small doubling type. We investigate the structure of a finitely generated group $G$ such that, for any set $S$ of generators of $G$ of minimal order, we have $S^2 leq 3|S|-beta$, where $beta in {1, 2, 3}$
{"title":"Groups with numerical restrictions on minimal generating sets","authors":"L. A. Kurdachenko, P. Longobardi, M. Maj","doi":"10.22108/IJGT.2019.115131.1526","DOIUrl":"https://doi.org/10.22108/IJGT.2019.115131.1526","url":null,"abstract":"We study an inverse problem of small doubling type. We investigate the structure of a finitely generated group $G$ such that, for any set $S$ of generators of $G$ of minimal order, we have $S^2 leq 3|S|-beta$, where $beta in {1, 2, 3}$","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"95-111"},"PeriodicalIF":0.2,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43007003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-13DOI: 10.22108/IJGT.2018.108082.1457
S. Ghoraishi
In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $Phi(G)$ has order $leq p^5$ admits a noninner automorphism of order $p$ leaving the center $Z(G)$ elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least $p^8$.
{"title":"On noninner automorphisms of finite $p$-groups that fix the center elementwise","authors":"S. Ghoraishi","doi":"10.22108/IJGT.2018.108082.1457","DOIUrl":"https://doi.org/10.22108/IJGT.2018.108082.1457","url":null,"abstract":"In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $Phi(G)$ has order $leq p^5$ admits a noninner automorphism of order $p$ leaving the center $Z(G)$ elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least $p^8$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"1-9"},"PeriodicalIF":0.2,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43953016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-09DOI: 10.22108/ijgt.2018.112124.1487
I. Subbotin, L. A. Kurdachenko, N. N. Semko
A subgroup H of a group G is called malonormal in G if H ∩H = ⟨1⟩ for every element x / ∈ NG(H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every selfnormalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.
{"title":"On some generalization of the malnormal subgroups","authors":"I. Subbotin, L. A. Kurdachenko, N. N. Semko","doi":"10.22108/ijgt.2018.112124.1487","DOIUrl":"https://doi.org/10.22108/ijgt.2018.112124.1487","url":null,"abstract":"A subgroup H of a group G is called malonormal in G if H ∩H = ⟨1⟩ for every element x / ∈ NG(H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every selfnormalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42675004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-09DOI: 10.22108/IJGT.2018.113482.1510
S. Stonehewer
Generalizing the concept of quasinormality, a subgroup $H$ of a group $G$ is said to be 4-quasinormal in $G$ if, for all cyclic subgroups $K$ of $G$, $langle H,Krangle=HKHK$. An intermediate concept would be 3-quasinormality, but in finite $p$-groups - our main concern - this is equivalent to quasinormality. Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups, particularly in finite $p$-groups. However, even in the smallest case, when $H$ is a 4-quasinormal subgroup of order $p$ in a finite $p$-group $G$, precisely how $H$ is embedded in $G$ is not immediately obvious. Here we consider one of these questions regarding the commutator subgroup $[H,G]$.
{"title":"$4$-quasinormal subgroups of prime order","authors":"S. Stonehewer","doi":"10.22108/IJGT.2018.113482.1510","DOIUrl":"https://doi.org/10.22108/IJGT.2018.113482.1510","url":null,"abstract":"Generalizing the concept of quasinormality, a subgroup $H$ of a group $G$ is said to be 4-quasinormal in $G$ if, for all cyclic subgroups $K$ of $G$, $langle H,Krangle=HKHK$. An intermediate concept would be 3-quasinormality, but in finite $p$-groups - our main concern - this is equivalent to quasinormality. Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups, particularly in finite $p$-groups. However, even in the smallest case, when $H$ is a 4-quasinormal subgroup of order $p$ in a finite $p$-group $G$, precisely how $H$ is embedded in $G$ is not immediately obvious. Here we consider one of these questions regarding the commutator subgroup $[H,G]$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"25-30"},"PeriodicalIF":0.2,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48755768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-01DOI: 10.22108/IJGT.2017.21478
E. Timoshenko
The Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_Gamma$ in a group of matrices $M_Gamma$. Properties and the universal theory of the group $M_Gamma$ are studied.
{"title":"On embedding of partially commutative metabelian groups to matrix groups","authors":"E. Timoshenko","doi":"10.22108/IJGT.2017.21478","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21478","url":null,"abstract":"The Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_Gamma$ in a group of matrices $M_Gamma$. Properties and the universal theory of the group $M_Gamma$ are studied.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"17-26"},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43043454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-01DOI: 10.22108/IJGT.2017.21476
R. Hafezieh
Let $G$ be a finite group. The prime degree graph of $G$, denoted by $Delta(G)$, is an undirected graph whose vertex set is $rho(G)$ and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible character degree of $G$. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this paper we consider the case where for a nonsolvable group $G$, $Delta(G)$ is a connected graph which has only one triangle and four vertices.
{"title":"On nonsolvable groups whose prime degree graphs have four vertices and one triangle","authors":"R. Hafezieh","doi":"10.22108/IJGT.2017.21476","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21476","url":null,"abstract":"Let $G$ be a finite group. The prime degree graph of $G$, denoted by $Delta(G)$, is an undirected graph whose vertex set is $rho(G)$ and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible character degree of $G$. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this paper we consider the case where for a nonsolvable group $G$, $Delta(G)$ is a connected graph which has only one triangle and four vertices.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"1-6"},"PeriodicalIF":0.2,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47541235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-01DOI: 10.22108/IJGT.2017.100688.1402
D. Malinin
Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let $K$ be a finite extension of the rational number field and $O_K$ the ring of integers of $K$. Let $G$ be a finite subgroup of $GL(2,K)$, the group of $(2 times 2)$-matrices over $K$. We obtain some conditions on $K$ for $G$ to be conjugate to a subgroup of $GL(2,O_K)$.
{"title":"On some integral representations of groups and global irreducibility.","authors":"D. Malinin","doi":"10.22108/IJGT.2017.100688.1402","DOIUrl":"https://doi.org/10.22108/IJGT.2017.100688.1402","url":null,"abstract":"Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let $K$ be a \u001cfinite extension of the rational number \u001cfield and $O_K$ the ring of integers of $K$. Let $G$ be a \u001cfinite subgroup of $GL(2,K)$, the group of $(2 times 2)$-matrices over $K$. We obtain some conditions on $K$ for $G$ to be conjugate to a subgroup of $GL(2,O_K)$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"81-94"},"PeriodicalIF":0.2,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47608354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-01DOI: 10.22108/IJGT.2017.100358.1398
D. D’Angeli, E. Rodaro
We address the problem of finding examples of non-bireversible transducers defining free groups, we show examples of transducers with sink accessible from every state which generate free groups, and, in general, we link this problem to the non-existence of certain words with interesting combinatorial and geometrical properties that we call fragile words. By using this notion, we exhibit a series of transducers constructed from Cayley graphs of finite groups whose defined semigroups are free, and thus having exponential growth.
{"title":"Fragile words and Cayley type transducers","authors":"D. D’Angeli, E. Rodaro","doi":"10.22108/IJGT.2017.100358.1398","DOIUrl":"https://doi.org/10.22108/IJGT.2017.100358.1398","url":null,"abstract":"We address the problem of finding examples of non-bireversible transducers defining free groups, we show examples of transducers with sink accessible from every state which generate free groups, and, in general, we link this problem to the non-existence of certain words with interesting combinatorial and geometrical properties that we call fragile words. By using this notion, we exhibit a series of transducers constructed from Cayley graphs of finite groups whose defined semigroups are free, and thus having exponential growth.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"95-109"},"PeriodicalIF":0.2,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47137389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-01DOI: 10.22108/IJGT.2017.21483
A. V. D. Luca, R. Ialenti
In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated.
研究了无限秩子群满足适当置换条件的非周期广义无限秩根群的结构。
{"title":"Groups with permutability conditions for subgroups of infinite rank","authors":"A. V. D. Luca, R. Ialenti","doi":"10.22108/IJGT.2017.21483","DOIUrl":"https://doi.org/10.22108/IJGT.2017.21483","url":null,"abstract":"In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"7 1","pages":"7-16"},"PeriodicalIF":0.2,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45754082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-13DOI: 10.22108/IJGT.2018.112574.1498
M. Vaughan-Lee
Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments.
{"title":"Graham Higman's PORC theorem","authors":"M. Vaughan-Lee","doi":"10.22108/IJGT.2018.112574.1498","DOIUrl":"https://doi.org/10.22108/IJGT.2018.112574.1498","url":null,"abstract":"Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"8 1","pages":"11-28"},"PeriodicalIF":0.2,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45248140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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