On Finite Group With Its Order Of 2-sylow Subgroup And The Largest And Second Largest Element Orders The Same As Mathieu Group

Let ?e(G)denote the set of orders of elements in G,K1(G)denote the largest element order of G,K2(G)denote the second largest element order of G.In the research of the structure of finite groups,it is not difficult to find that some quanti-tative properties of groups greatly determine the structure of groups,such as Sylow theorem,Odd order solvable theorem,etc.In the 1980s,professor Wujie Shi conjec-tured a new characterization of a simple group by using two quantities:the order of G and ?e(G).After the conjecture was put forward,many scholars have discussed it.Professor Wujie Shi has also done a lot of work and proved a new characterization of almost all finite simple groups by using two quantities:the order of G and ?e(G).In 2009,Russian mathematician Vasilev A V,et al,proved the conjecture.It is concluded that:Let G be a finite group,M be a finite simple group,then G?M if and only if|G|=|M|and ?e(G)=?e(M).After the conjecture was finished,scholars tried to characterize simple groups by using less quantities.Professor Guiyun Chen and Liguan He characterized some alternating groups and some symmetric groups by using the order of G and the largest and second largest element orders of G.It is concluded that:(1)Let G be a finite group,M be alternating group An(n=5,6,7,9,10,11,13),then G?M if and only if |G|=|M| and K1(G)=K1(M).(2)Let G be a finite group,M be alternating group An(n=8,12),then G?M if and only if |G|=|M| and Ki(G)=Ki(M),where i=1,2.(3)Let G be a finite group,M be symmetric group S5,S6,S7,then G?M if and only if Ki(G)=K1(M)and |G|=|M|.However,in similar researches,the order of G is regarded as a necessary con-dition.Prof'essor Guiyun Chen and Meng Chen replaced the order of G with the order of its 2-Sylow subgroup to discuss groups.They discussed the structure of finite groups whose the largest element order is 5 and whose the order of the Sylow 2-subgroup is 2,4,8.And they researched finite groups whose the largest element order is 7 and whose the order of Sylow 2-subgroup is 8.Although simple groups cannot be characterized when only the order of 2-Sylow subgroup is restricted,this study is of theoretical significance to observe the changes of group structure under different quantities.In this paper,the structure of finite groups having the same order of 2-Sylow subgroup and the same largest and second largest element orders with Mathieu group(M11,M12,M22,M23,M24)has been discussed.The some properties of order of 2-Sylow subgroup,outer automorphism groups,prime graph components and its isolated points of groups are used in discussion.It is concluded that:Theorem 3.1 Let G have the same order of 2-Sylow subgroup with M11 and K1(G)=K1(M11),K2(G)=K2(M11),then:(1)G/H?H11,|G|=24·3a·5·11,where H is a nilpotent group,a?2,|H|=3a-2,and exp(H)|3;(2)G=HK where K is the Frobenius kernel and H is the Frobenius comple-ment,|G|=24·3a·11d,H2 is a generalized quaternion group,H3 is a cyclic group,and K is an elementary Abel 11-group;(3)G=HK where K is the Frobenius kernel and H is the Frobenius comple-ment,|G|=24·11d,H is a generalized quaternion group,and K is an elementary Abel 11-group.Theorem 3.2 Let G have the same order of 2-Sylow subgroup with M12 and K1(G)=K1(M12),K2(G)=K2(M12),then:G/H?M12,|G|=26·3a·5b·11,where H is a nilpotent group,a?3,b?1,|H|=3a-3 or 5b-1,and exp(H)|9 or exp(H)=5.Theorem 3.3 Let G have the same order of 2-Sylow subgroup with M22 and K1(G)=K1(M22),K2(G)=K2(M22),then:G/H?M22,|G|=27·3a·5·7·11,where H is a nilpotent group,a?2,|H|=3a-2,and exp(H)|3.Theorem 3.4 Let G have the same order of 2-Sylow subgroup with M23 and K1(G)=K1(M23),K2(G)=K2(M23),then:(1)G/H?M23,|G|=27·3a·5b·7·11·23,where H is a nilpotent group,a?2,b?1,|H|=3a-2 or 5b-1,and exp(H)|9 or exp(H)=5;(2)G/H? M23,|G|=27·3a·5·7c·11·23,where H is a nilpotent group,a?2,c?1,|H|=3a-2 or 7c-1,and exp(H)|9 or exp(H)=7;(3)G/H? M23,|G|=27·32·5b·7c·11·23,where H is a nilpotent group,b?1,c?1,|H|=5b-1 or 7c-1,and exp(H)=5 or exp(H)=7.Theorem 3.5 Let G have the same order of 2-Sylow subgroup with M24 and K1(G)=K1(M24),K2(G)=K2(M24),then:(1)G/H? M24,|G|=210·3a·5b·7·11·23,where H is a nilpotent group,a? 3,b?1,|H|=3a-3 or 5b-1,and exp(H)|9 or exp(H)=5;(2)G/H? M24,|G|=210·3a·5·7c·11·23,where H is a nilpotent group,a?3,c? 1,|H|=3a-3 or 7c-1,and exp(H)|9 or exp(H)=7;(3)G/H? H24,|G|=210·33·5b·7c·11·23,where H is a nilpotent group,b?1,c?1,|H|=5b-1 or 7c-1,and exp(H)=5 or exp(H)=7.