{"title":"Finite p-groups with cyclic center have non-inner automorphisms of order p","authors":"Mandeep Singh, Mahak Sharma","doi":"10.1007/s00013-025-02112-2","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>p</i> be a prime number. A longstanding conjecture asserts that every finite non-abelian <i>p</i>-group has a non-inner automorphism of order <i>p</i>. In this paper, under some conditions on an odd order finite <i>p</i>-group <i>G</i> with cyclic center, we prove that <i>G</i> exhibits a non-inner automorphism of order <i>p</i>. As a consequence, under certain conditions on a finite <i>p</i>-group <i>G</i> <span>\\((p>2),\\)</span> the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3, which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 5","pages":"469 - 474"},"PeriodicalIF":0.5000,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-025-02112-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let p be a prime number. A longstanding conjecture asserts that every finite non-abelian p-group has a non-inner automorphism of order p. In this paper, under some conditions on an odd order finite p-group G with cyclic center, we prove that G exhibits a non-inner automorphism of order p. As a consequence, under certain conditions on a finite p-group G\((p>2),\) the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3, which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).
设p是质数。一个长期猜想证明了每一个有限非阿贝p群都有p阶的非内自同构。本文在具有循环中心的奇阶有限p群G的某些条件下,证明了G具有p阶的非内自同构。因此,在有限p群G的某些条件下\((p>2),\)证明了除了类2和极大类以外的所有幂零类的猜想。此外,我们还解决了一些非阿贝尔有限的共3类3群的猜想,这是Ruscitti et al. (Monatsh)的主要结果的一个待决情况。数学。183(4):679-697,2016)。
期刊介绍:
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.