Pub Date : 2021-06-24DOI: 10.22108/IJGT.2021.128359.1686
S. Pradhan, B. Sury
For a finite group $G$, three of the positive integers governing its representation theory over $mathbb{C}$ and over $mathbb{Q}$ are $p(G),q(G),c(G)$. Here, $p(G)$ denotes the {it minimal degree} of a faithful permutation representation of $G$. Also, $c(G)$ and $q(G)$ are, respectively, the minimal degrees of a faithful representation of $G$ by quasi-permutation matrices over the fields $mathbb{C}$ and $mathbb{Q}$. We have $c(G)leq q(G)leq p(G)$ and, in general, either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the {it holomorph} of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n geq 3$. We explicitly describe the set of all {it isomorphism types} of irreducible representations of $G$ over the field of complex numbers $mathbb{C}$ as well as the isomorphism types over the field of rational numbers $mathbb{Q}$. We compute the {it Wedderburn decomposition} of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $mathbb{C}$ and over $mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.
{"title":"Rational and Quasi-Permutation Representations of Holomorphs of Cyclic $p$-Groups","authors":"S. Pradhan, B. Sury","doi":"10.22108/IJGT.2021.128359.1686","DOIUrl":"https://doi.org/10.22108/IJGT.2021.128359.1686","url":null,"abstract":"For a finite group $G$, three of the positive integers governing its representation theory over $mathbb{C}$ and over $mathbb{Q}$ are $p(G),q(G),c(G)$. Here, $p(G)$ denotes the {it minimal degree} of a faithful permutation representation of $G$. Also, $c(G)$ and $q(G)$ are, respectively, the minimal degrees of a faithful representation of $G$ by quasi-permutation matrices over the fields $mathbb{C}$ and $mathbb{Q}$. We have $c(G)leq q(G)leq p(G)$ and, in general, either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the {it holomorph} of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n geq 3$. We explicitly describe the set of all {it isomorphism types} of irreducible representations of $G$ over the field of complex numbers $mathbb{C}$ as well as the isomorphism types over the field of rational numbers $mathbb{Q}$. We compute the {it Wedderburn decomposition} of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $mathbb{C}$ and over $mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48531336","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 : 2021-05-20DOI: 10.22108/IJGT.2021.125453.1652
M. Shahryari
We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize Theorem G of [J. Algebra, 219 (1999) 16--79]. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{omega}$-compact, they are geometrically equivalent.
{"title":"Nullstellensatz for relative existentially closed groups","authors":"M. Shahryari","doi":"10.22108/IJGT.2021.125453.1652","DOIUrl":"https://doi.org/10.22108/IJGT.2021.125453.1652","url":null,"abstract":"We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize Theorem G of [J. Algebra, 219 (1999) 16--79]. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{omega}$-compact, they are geometrically equivalent.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49270801","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 : 2021-04-05DOI: 10.22108/IJGT.2021.123705.1632
B. Rodrigues
A binary triply-even $[98280, 25, 47104]_2$ code invariant under the sporadic simple group ${rm Co}_1$ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of ${rm Co}_1$ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${rm Co}_1$. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.
{"title":"On some projective triply-even binary codes invariant under the Conway group ${rm Co}_1$","authors":"B. Rodrigues","doi":"10.22108/IJGT.2021.123705.1632","DOIUrl":"https://doi.org/10.22108/IJGT.2021.123705.1632","url":null,"abstract":"A binary triply-even $[98280, 25, 47104]_2$ code invariant under the sporadic simple group ${rm Co}_1$ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of ${rm Co}_1$ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${rm Co}_1$. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"14 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205302","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 : 2021-02-22DOI: 10.22108/IJGT.2021.127679.1681
P. Cameron
This paper concerns aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action of the automorphism group of $G$). The particular graphs I will chiefly discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph. My main concern is not with properties of these graphs individually, but rather with comparisons between them. The graphs mentioned, together with the null and complete graphs, form a hierarchy (as long as $G$ is non-abelian), in the sense that the edge set of any one is contained in that of the next; interesting questions involve when two graphs in the hierarchy are equal, or what properties the difference between them has. I also consider various properties such as universality and forbidden subgraphs, comparing how these properties play out in the different graphs. I have also included some results on intersection graphs of subgroups of various types, which are often in a ``dual'' relation to one of the other graphs considered. Another actor is the Gruenberg--Kegel graph, or prime graph, of a group: this very small graph has a surprising influence over various graphs defined on the group. Other graphs which have been proposed, such as the nilpotence, solvability, and Engel graphs, will be touched on rather more briefly. My emphasis is on finite groups but there is a short section on results for infinite groups. There are briefer discussions of general $Aut(G)$-invariant graphs, and structures other than groups (such as semigroups and rings). Proofs, or proof sketches, of known results have been included where possible. Also, many open questions are stated, in the hope of stimulating further investigation.
{"title":"Graphs defined on groups","authors":"P. Cameron","doi":"10.22108/IJGT.2021.127679.1681","DOIUrl":"https://doi.org/10.22108/IJGT.2021.127679.1681","url":null,"abstract":"This paper concerns aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action of the automorphism group of $G$). The particular graphs I will chiefly discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph. \u0000 My main concern is not with properties of these graphs individually, but rather with comparisons between them. The graphs mentioned, together with the null and complete graphs, form a hierarchy (as long as $G$ is non-abelian), in the sense that the edge set of any one is contained in that of the next; interesting questions involve when two graphs in the hierarchy are equal, or what properties the difference between them has. I also consider various properties such as universality and forbidden subgraphs, comparing how these properties play out in the different graphs. \u0000 I have also included some results on intersection graphs of subgroups of various types, which are often in a ``dual'' relation to one of the other graphs considered. Another actor is the Gruenberg--Kegel graph, or prime graph, of a group: this very small graph has a surprising influence over various graphs defined on the group. \u0000 Other graphs which have been proposed, such as the nilpotence, solvability, and Engel graphs, will be touched on rather more briefly. My emphasis is on finite groups but there is a short section on results for infinite groups. There are briefer discussions of general $Aut(G)$-invariant graphs, and structures other than groups (such as semigroups and rings). \u0000Proofs, or proof sketches, of known results have been included where possible. Also, many open questions are stated, in the hope of stimulating further investigation.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49533473","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 : 2021-01-27DOI: 10.22108/IJGT.2021.127210.1673
Shawn T. Burkett, M. Lewis
In this paper, we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients. This characterization is based on counting formulas due to Gallagher.
{"title":"A characterization of GVZ groups in terms of fully ramified characters","authors":"Shawn T. Burkett, M. Lewis","doi":"10.22108/IJGT.2021.127210.1673","DOIUrl":"https://doi.org/10.22108/IJGT.2021.127210.1673","url":null,"abstract":"In this paper, we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients. This characterization is based on counting formulas due to Gallagher.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41490471","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 : 2021-01-11DOI: 10.22108/IJGT.2021.119299.1575
A. Trofimuk
A subgroup $A$ of a group $G$ is called {it seminormal} in $G$, if there exists a subgroup $B$ such that $G=AB$ and $AX$~is a subgroup of $G$ for every subgroup $X$ of $B$. The group $G = G_1 G_2 cdots G_n$ with pairwise permutable subgroups $G_1,ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i, jin {1,ldots,n}$, $ineq j$, is studied. In particular, we prove that if $G_iin frak F$ for all $i$, then $G^frak Fleq (G^prime)^frak N$, where $frak F$ is a saturated formation and $frak U subseteq frak F$. Here $frak N$ and $frak U$~ are the formations of all nilpotent and supersoluble groups respectively, the $mathfrak F$-residual $G^frak F$ of $G$ is the intersection of all those normal subgroups $N$ of $G$ for which $G/N in mathfrak F$.
{"title":"A note on groups with a finite number of pairwise permutable seminormal subgroups","authors":"A. Trofimuk","doi":"10.22108/IJGT.2021.119299.1575","DOIUrl":"https://doi.org/10.22108/IJGT.2021.119299.1575","url":null,"abstract":"A subgroup $A$ of a group $G$ is called {it seminormal} in $G$, if there exists a subgroup $B$ such that $G=AB$ and $AX$~is a subgroup of $G$ for every subgroup $X$ of $B$. The group $G = G_1 G_2 cdots G_n$ with pairwise permutable subgroups $G_1,ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i, jin {1,ldots,n}$, $ineq j$, is studied. In particular, we prove that if $G_iin frak F$ for all $i$, then $G^frak Fleq (G^prime)^frak N$, where $frak F$ is a saturated formation and $frak U subseteq frak F$. Here $frak N$ and $frak U$~ are the formations of all nilpotent and supersoluble groups respectively, the $mathfrak F$-residual $G^frak F$ of $G$ is the intersection of all those normal subgroups $N$ of $G$ for which $G/N in mathfrak F$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47990075","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 : 2021-01-01DOI: 10.22108/IJGT.2020.122990.1622
Gholamreza Rafatneshan, Y. Zamani
Let $V$ be a unitary space. Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$. Consider the generalized symmetrizer on the tensor space $Uotimes V^{otimes m}$, $$ S_{Lambda}(uotimes v^{otimes})=dfrac{1}{|G|}sum_{sigmain G}Lambda(sigma)uotimes v_{sigma^{-1}(1)}otimescdotsotimes v_{sigma^{-1}(m)} $$ defined by $G$ and $Lambda$. The image of $Uotimes V^{otimes m}$ under the map $S_Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $Lambda$ and is denoted by $V_Lambda(G)$. The elements in $V_Lambda(G)$ of the form $S_{Lambda}(uotimes v^{otimes})$ are called generalized decomposable tensors and are denoted by $ucircledast v^{circledast}$. For any linear operator $T$ acting on $V$, there is a unique induced operator $K_{Lambda}(T)$ acting on $V_{Lambda}(G)$ satisfying $$ K_{Lambda}(T)(uotimes v^{otimes})=ucircledast Tv_{1}circledast cdots circledast Tv_{m}. $$ If $dim U=1$, then $K_{Lambda}(T)$ reduces to $K_{lambda}(T)$, induced operator on symmetry class of tensors $V_{lambda}(G)$. In this paper, the basic properties of the induced operator $K_{Lambda}(T)$ are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions.
{"title":"Induced operators on the generalized symmetry classes of tensors","authors":"Gholamreza Rafatneshan, Y. Zamani","doi":"10.22108/IJGT.2020.122990.1622","DOIUrl":"https://doi.org/10.22108/IJGT.2020.122990.1622","url":null,"abstract":"Let $V$ be a unitary space. Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$. Consider the generalized symmetrizer on the tensor space $Uotimes V^{otimes m}$, $$ S_{Lambda}(uotimes v^{otimes})=dfrac{1}{|G|}sum_{sigmain G}Lambda(sigma)uotimes v_{sigma^{-1}(1)}otimescdotsotimes v_{sigma^{-1}(m)} $$ defined by $G$ and $Lambda$. The image of $Uotimes V^{otimes m}$ under the map $S_Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $Lambda$ and is denoted by $V_Lambda(G)$. The elements in $V_Lambda(G)$ of the form $S_{Lambda}(uotimes v^{otimes})$ are called generalized decomposable tensors and are denoted by $ucircledast v^{circledast}$. For any linear operator $T$ acting on $V$, there is a unique induced operator $K_{Lambda}(T)$ acting on $V_{Lambda}(G)$ satisfying $$ K_{Lambda}(T)(uotimes v^{otimes})=ucircledast Tv_{1}circledast cdots circledast Tv_{m}. $$ If $dim U=1$, then $K_{Lambda}(T)$ reduces to $K_{lambda}(T)$, induced operator on symmetry class of tensors $V_{lambda}(G)$. In this paper, the basic properties of the induced operator $K_{Lambda}(T)$ are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"10 1","pages":"197-211"},"PeriodicalIF":0.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68205274","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 : 2020-12-01DOI: 10.22108/IJGT.2020.120132.1584
A. Tortora, M. Tota
The eighth edition of the international series of Groups St Andrews conferences was held at the University of Bath in 2009 and one of the theme days was dedicated to Engel groups. Since then much attention has been devoted to a verbal generalization of Engel groups. In this paper we will survey the development of this investigation during the last decade.
{"title":"Engel groups in bath - ten years later","authors":"A. Tortora, M. Tota","doi":"10.22108/IJGT.2020.120132.1584","DOIUrl":"https://doi.org/10.22108/IJGT.2020.120132.1584","url":null,"abstract":"The eighth edition of the international series of Groups St Andrews conferences was held at the University of Bath in 2009 and one of the theme days was dedicated to Engel groups. Since then much attention has been devoted to a verbal generalization of Engel groups. In this paper we will survey the development of this investigation during the last decade.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"251-260"},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43010193","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 : 2020-12-01DOI: 10.22108/IJGT.2020.119749.1581
O. Puglisi, G. Traustason
An automorphism $alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_nalpha]=1$ for all $gin G$. In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups. We also show that, assuming the truth of a conjecture about the representation theory of solvable groups raised by P. Neumann, it is possible to produce, for a suitable prime $p$, an example of a f.g. solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group. Conversely we show that, if there exists a f.g. solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms, then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent.
{"title":"Some remarks on unipotent automorphisms","authors":"O. Puglisi, G. Traustason","doi":"10.22108/IJGT.2020.119749.1581","DOIUrl":"https://doi.org/10.22108/IJGT.2020.119749.1581","url":null,"abstract":"An automorphism $alpha$ of the group $G$ is said to be $n$-unipotent if $[g,_nalpha]=1$ for all $gin G$. In this paper we obtain some results related to nilpotency of groups of $n$-unipotent automorphisms of solvable groups. We also show that, assuming the truth of a conjecture about the representation theory of solvable groups raised by P. Neumann, it is possible to produce, for a suitable prime $p$, an example of a f.g. solvable group possessing a group of $p$-unipotent automorphisms which is isomorphic to an infinite Burnside group. Conversely we show that, if there exists a f.g. solvable group $G$ with a non nilpotent $p$-group $H$ of $n$-automorphisms, then there is such a counterexample where $n$ is a prime power and $H$ has finite exponent.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"293-300"},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47965125","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 : 2020-12-01DOI: 10.22108/IJGT.2020.119870.1582
S. Hart, Daniel McVeagh
Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $pth$ roots $g$ can have? We write $myro_p(G)$, or just $myro_p$, for the maximum value of $frac{1}{|G|}|{x in G: x^p=g}|$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $myro_p$ is large. If there is an element $g$ of $G$ with more $pth$ roots than the identity, then we show $myro_p(G) leq myro_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $myro_p(G) leq frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $myro_p(G) leq frac{1}{p} + frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $myro_2$, and give partial results on groups with high values of $myro_3$.
{"title":"Groups with many roots","authors":"S. Hart, Daniel McVeagh","doi":"10.22108/IJGT.2020.119870.1582","DOIUrl":"https://doi.org/10.22108/IJGT.2020.119870.1582","url":null,"abstract":"Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $pth$ roots $g$ can have? We write $myro_p(G)$, or just $myro_p$, for the maximum value of $frac{1}{|G|}|{x in G: x^p=g}|$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $myro_p$ is large. If there is an element $g$ of $G$ with more $pth$ roots than the identity, then we show $myro_p(G) leq myro_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $myro_p(G) leq frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $myro_p(G) leq frac{1}{p} + frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $myro_2$, and give partial results on groups with high values of $myro_3$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":"9 1","pages":"261-276"},"PeriodicalIF":0.2,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42821005","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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