关于部分单K_4-群的自同构群的刻画
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摘要
【目的】为了弱化有限群数量刻画的数量条件。【方法】用群的阶,最高阶元素的阶及次高阶元素的阶刻画单K4-群的自同构群。【结果】证明了A7,A9,G2(3),U3(4),U3(9),3 D4(2),S4(4),L3(8),U3(7),A10,M11,M12,J2,Sz(8),Sz(32)和S6(2)的自同构群可以由群的阶,最高阶元素的阶唯一刻画,而A8,U5(2)和L3(5)的自同构群可以由群的阶,最高阶元素的阶及次高阶元素的阶唯一刻画。【结论】结果说明上述单K4-群的自同构群最多需要3个数量就可以唯一决定。
[Purposes]It aims to find fewer numbers to characterize a finite group.[Methods]The automorphism groups of simple K4-groups by using group order is characterized,the largest element order and the second largest element order.[Findings]The automorphism groups of A7,A9,G2(3),U3(4),U3(9),3 D4(2),S4(4),L3(8),U3(7),A10,M11,M12,J2,Sz(8),Sz(32)and S6(2)can be characterized by using group order and the largest element order.The automorphism groups of A8,U5(2)and L3(5)can be characterized by using group order,the largest element order and the second largest element order.[Conclusions]The present study shows that the automorphism groups of above simple K4-groups can be uniquely determined by at most three numbers.
[Purposes]It aims to find fewer numbers to characterize a finite group.[Methods]The automorphism groups of simple K4-groups by using group order is characterized,the largest element order and the second largest element order.[Findings]The automorphism groups of A7,A9,G2(3),U3(4),U3(9),3 D4(2),S4(4),L3(8),U3(7),A10,M11,M12,J2,Sz(8),Sz(32)and S6(2)can be characterized by using group order and the largest element order.The automorphism groups of A8,U5(2)and L3(5)can be characterized by using group order,the largest element order and the second largest element order.[Conclusions]The present study shows that the automorphism groups of above simple K4-groups can be uniquely determined by at most three numbers.
引文
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[13]何立官,陈贵云.关于3′-单K4-群及其自同构群的新刻画[J].山西大学学报(自然科学版),2013,36(4):540-543.HE L G,CHEN G Y.A new characterization of 3′-simple K4-groups and their automorphism groups[J].Journal of Shanxi University(Natural Science),2013,36(4):540-543.
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[15]HE L G,CHEN G Y,XU H J.A new characterization of sporadic simple groups[J].Italian Journal of Pure and Applied Mathematics,2013,30:373-392.
[16]伍星,马玉龙,刘海林.自同构群的阶等于其阶两倍的有限p-群[J].重庆理工大学学报(自然科学),2017,31(12):189-191.WU X,MA Y L,LIU H L.A finite p-group whose order of its automorphism group is equal to twice its order[J].Journal of Chongqing University of Technology(Natural Science),2017,31(12):189-191.
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[19]陈贵云.关于Frobenius群和2-Frobenius群[J].西南师范大学学报(自然科学版),1995,20(5):485-487.CHEN G Y.On structure of Frobenius group and 2-Frobenius group[J].Journal of Southwest China Normal University(Natural Science),1995,20(5):485-487.
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[2]SHI W J,BI J X.A characterization of the alternating groups[J].Southeast Asian Bulletin of Mathematics,1962,16(1):81-90.
[3]SHI W J,BI J X.A characterization of Suzuki-Ree-Groups[J].Science in China,Ser A,1991,34(1):14-19.
[4]SHI W J,BI J X.A characteristic property for each finite projective special linear groups[J].Lecture Notes in Math,1990,1456:171-180.
[5]SHI W J.Pure quantitative characterization of finite groups[J].Progressin Nature Science,1994,4(3):316-326.
[6]CAO H P,SHI W J.Pure quantitative characterization of finite projective special unitary groups[J].Science in China,Ser A,2002,45:761-772.
[7]XU M C,SHI W J.Pure quantitative characterization of finite simple groups 2 Dn(q)and Dl(q)(l odd)[J].Algebra Colloquium,2003,10:427-443.
[8]VASIL’EV A V,GRECHKOSEEVA M A,MAZUROV V D.Characterization of the finite simple groups by spectrum and order[J].Algebra and Logic,2009,48(6):385-409.
[9]HE L G,CHEN G Y.A new characterization of simple K3-groups[J].Communications in Algebra,2012,40(10):3903-3911.
[10]何立官,徐海静.关于单K3-群的自同构群的刻画[J].数学进展,2015,44(3):363-368.HE L G,XU H J.A characterization of automorphism groups of simple K3-groups[J].Advances in Mathematics(CHINA),2015,44(3):363-368.
[11]何立官,陈贵云.关于型为L2(p)的单K4-群的一个新刻画[J].数学进展,2014,43(5):667-670.HE L G,CHEN G Y.A new characterization of simple K4-groups with type L2(p)[J].Advances in Mathematics(CHINA),2014,43(5):667-670.
[12]高彦伟,曹洪平.关于散在单群的自同构群的一个新刻画[J].西南师范大学学报(自然科学版),2013,38(2):1-5.GAO Y W,CAO H P.On a new characterization of the automorphism groups in sporadic simple groups[J].Journal of Southwest China Normal University(Natural Science),2013,38(2):1-5.
[13]何立官,陈贵云.关于3′-单K4-群及其自同构群的新刻画[J].山西大学学报(自然科学版),2013,36(4):540-543.HE L G,CHEN G Y.A new characterization of 3′-simple K4-groups and their automorphism groups[J].Journal of Shanxi University(Natural Science),2013,36(4):540-543.
[14]HE L G,CHEN G Y.A new characterization simple K4-groups[J].Journal of Mathematical Research with Applications,2015,35(4):400-406.
[15]HE L G,CHEN G Y,XU H J.A new characterization of sporadic simple groups[J].Italian Journal of Pure and Applied Mathematics,2013,30:373-392.
[16]伍星,马玉龙,刘海林.自同构群的阶等于其阶两倍的有限p-群[J].重庆理工大学学报(自然科学),2017,31(12):189-191.WU X,MA Y L,LIU H L.A finite p-group whose order of its automorphism group is equal to twice its order[J].Journal of Chongqing University of Technology(Natural Science),2017,31(12):189-191.
[17]WILLIAMS J S.Prime graph components of finite group[J].Algebra,1981,69:487-513.
[18]BUGEAUD Y,CAO Z,MIGNOTTE M.On simple K4-Groups[J].Algebra,2001,241:658-668.
[19]陈贵云.关于Frobenius群和2-Frobenius群[J].西南师范大学学报(自然科学版),1995,20(5):485-487.CHEN G Y.On structure of Frobenius group and 2-Frobenius group[J].Journal of Southwest China Normal University(Natural Science),1995,20(5):485-487.
[20]GORENSTEIN D.Finite Groups[M].New York:Chelsea Publishing Company,1980.
[21]CONWAY J H,CURTIS R T,NORTON S P,et al.Atlas of Finite Groups[M].Oxford:Clarendon Press,1985.