JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (11): 20-26.doi: 10.6040/j.issn.1671-9352.0.2023.258

Previous Articles    

A new characterization of A5 and PSL(2,7)

KUANG Meiqun, LU Jiakuan, LI Yu, ZHANH Boru*   

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, Guangxi, China
  • Published:2025-11-11

PDF (PC)

54

Abstract: Let G be a finite group, and let m(G)=∑g∈G1/(o(g)), where o(g) denotes the order of g∈G, and h(G) is the maximal order of the elements in G. We give a new characterization of A5 and PSL(2,7), that is, G≅A5 if and only if m(G)=m(A5) and h(G)=h(A5). G≅ PSL(2,7)if and only if m(G)=m(PSL(2,7))and h(G)=h(PSL(2,7)).

Key words: the order of elements, alternating group, simple group, sylow subgroup, finite group

CLC Number: 

  • O152.1
[1] 施武杰. A5的一个特征性质[J]. 西南师范大学学报(自然科学版),1986,11(3):11-14. SHI Wujie. A characteristic property of A5[J]. Journal of Southwest University:(Natural Science), 1986, 11(3):11-14.
[2] 郭继东,任永才,张志让. 单群 PSL(2,7)的一个新刻画[J]. 四川大学学报(自然科学版),2014,51(4):696-700. GUO Jidong, REN Yongcai, ZHANG Zhirang. A new characterization of cingle group PSL(2,7)[J]. Journal of Sichuan University:(Natural Science), 2014, 51(4):696-700.
[3] JAFARIAN AMIRI S M. Characterization of A5 and PSL(2,7)by sum of element orders[J]. International Journal of Group Theory, 2013, 2(2):35-39.
[4] 王华丽. 利用元素阶之和及最高阶刻画群[D]. 重庆:西南大学,2015. WANG Huali. Characterizing groups by the sum of element orders and the maximal order[D]. Chongqing: Southwest University, 2015.
[5] AMIRI H, JAFARIAN AMIRI S M, ISAACS I M. Sums of element orders in finite groups[J]. Communications in Algebra, 2009, 37(9):2978-2980.
[6] AZAD M B, KHOSRAVI B. A criterion for solvability of a finite group by the sum of element orders[J]. Journal of Algebra, 2018, 516:115-124.
[7] AZAD M B, KHOSRAVI B, RASHIDI H. On the sum of the inverses of the element orders in finite groups[J]. Communications in Algebra, 2023, 51(2): 694-698.
[8] AZAD M B, KHOSRAVI B. Properties of finite groups determined by the product of their element orders[J]. Bulletin of the Australian Mathematical Society, 2021, 103(1):88-95.
[9] TARNAUCEANU M. Detecting structural properties of finite groups by the sum of element orders[J]. Israel Journal of Mathematics, 2020, 238(2):629-637.
[10] AMIRI H, JAFARIAN AMIRI S M. Sum of element orders on finite groups of the same order[J]. Journal of Algebra and Its Applications, 2011, 10(2):187-190.
[11] 徐明曜. 有限群导引[M]. 北京:科学出版社,2001. XU Mingyao. Finite group guidance[M]. Beijing: Natural Science, 2001.
[1] JI Jiuzhou, ZHOU Wei, YANG Nanying. Influence of σ-permutably embedded subgroups on p-nilpotency of finite groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(5): 25-32.
[2] LIU Xin, WU Zhen-feng, YANG Nan-ying. Finite groups with some m-S-supplemented subgroups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(4): 8-13.
[3] ZHAO Le-le, HAI Jin-ke. Coleman automorphisms of extensions of the finite groups by some groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(10): 109-112.
[4] HAI Jin-ke, WANG Wei, HE Wei-ping. A note on Coleman automorphisms of finite groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 35-38.
[5] CHEN Song-liang. On the structures of groups of order p2q3 with non-Abelian Sylow subgroups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 93-97.
[6] CHEN Song-liang, LI Jing-lei, OUYANG Jian-xin. The structions of finite groups of order p3q [J]. J4, 2013, 48(2): 27-31.
[7] LIU Xiao-lei. A note on a theorem of E.Alemany [J]. J4, 2012, 47(10): 7-13.
[8] ZUO Jin-ming1, ZHANG Yao-ming1, ZHANG Tian-de2, LI Na2. The alternating group method for KuramotoSivashinsky equation [J]. J4, 2011, 46(12): 29-32.
[9] PU Zhao-Nian1, TANG Ju-Ping2, GU Chun-Hua3. The influence of the properties of local subgroups on the structure of finite groups [J]. J4, 2011, 46(12): 51-54.
[10] FU Ji-mei1, ZUO Jin-ming2*, ZHANG Tian-de1. The alternating group method for fifth-order dispersive equation [J]. J4, 2010, 45(6): 46-49.
[11] ZUO Jinming. An alternating group method for a fifthorder dispersive equation [J]. J4, 2009, 44(4): 79-83 .
[12] LIU Xiao-Lei, WANG Yan-Ming. q-supplement in finite groups and its applications [J]. J4, 2009, 44(10): 87-90.
[13] YU Gui-hai,QU Hui . The dimension of the principle indecomposable module for the finite group of type C2 [J]. J4, 2008, 43(8): 69-71 .
[14] ZHANG Qing-jie,WANG Wen-qia . A high order parallel iterative scheme for the dispersive equation [J]. J4, 2008, 43(2): 8-11 .
[15] LIU Xiao-lei . On generalized c-normality of finite groups [J]. J4, 2008, 43(10): 18-20 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd