{"title":"单词、置换和有限群的不可解长度","authors":"Alexander Bors, A. Shalev","doi":"10.4171/JCA/51","DOIUrl":null,"url":null,"abstract":"We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\\rightarrow G$ has a fiber of size at least $\\rho|G|^d$ for some fixed $\\rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $\\rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $\\rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Words, permutations, and the nonsolvable length of a finite group\",\"authors\":\"Alexander Bors, A. Shalev\",\"doi\":\"10.4171/JCA/51\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\\\\rightarrow G$ has a fiber of size at least $\\\\rho|G|^d$ for some fixed $\\\\rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $\\\\rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $\\\\rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.\",\"PeriodicalId\":48483,\"journal\":{\"name\":\"Journal of Combinatorial Algebra\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2019-04-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/JCA/51\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/JCA/51","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Words, permutations, and the nonsolvable length of a finite group
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\rightarrow G$ has a fiber of size at least $\rho|G|^d$ for some fixed $\rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $\rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $\rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.