Finite Groups Having Their Order Of 2-Sylow Subgroup And The Largest And Second Largest Element Order The Same As Those Of A₉ Or A₁₀

The paper discussed the structure of finite groups having their order of 2-Sylow subgroup, and the largest and second largest orders the same as those of Ag or A10 is given.Firstly,we research the structure of finite groups having their order of 2-Sylow subgroup, and the largest and second largest orders the same as those of A9 is given,the following theorem is obtained.Theorem 4.1 Let G be a finite group which the order of 2-Sylow subgroup is 26,the order of the highest order element is 15,the order of the next highest order element is 12,then G is {2,3,5}-group or the following One is established:(1) G/H≌L2(7),H is a nilpotent group of exponent dividing 10 and of the order is 23·5β.(2) G/H≌L2(8),H is a nilpotent group of exponent dividing 10 and of the order is 23·5β.(3) G/H≌L2(8)·3,H is a nilpotent group of exponent dividing 10 and of the order is 23·5β.(4) G/H≌U3(3)·2,H is a nilpotent group of exponent is 5 and of the order is 5β or H is a nilpotent group of exponent dividing 15 and of the order is 3α-3·5β.(5) G/H≌A7, H is a nilpotent group of exponent dividing 8 and of the order is 23 or H is a nilpotent group of exponent dividing 12 and of the order is 23·3α-2 or H is a nilpotent group of exponent dividing 10 and of the order is 23·5β-1.(6) G/H≌A8,is a nilpotent group of exponent dividing 9 and of the order is 3α-2 or H is a nilpotent group of exponent is 5 and of the order is 5β-1 or H is a nilpotent group of exponent dividing 15 and of the order is 3α-2·5β-1.(7) G/H≌L3(4),H is a nilpotent group of exponent dividing 9 and of the order is 3α- or H is a nilpotent group of exponent is 5 and of the order is 5β-1 or H is a nilpotent group of exponent dividing 15 and of the order is 3α-2·5β-1.(8) G/H≌A9,H is a nilpotent group of exponent dividing 9 and of the order is 3α-4 or H is a nilpotent group of exponent is 5 and of the order is 5β-1 or H is a nilpotent group of exponent dividing 15 and of the order is 3α-4·5β-1.(9) G/H≌M12,H is a nilpotent group of exponent dividing 9 and of the order is 3α-3 or H is a nilpotent group of exponent is 5 and of the order is 5β-1 or H is a nilpotent group of exponent dividing 15 and of the order is 3α-3·5β-1.Secondly,we discuss the structure of finite groups having their order of 2-Sylow subgroup,and the largest and second largest orders the same as those of A10 is given,the following theorem is obtained.Theorem 4.2 Let G be a finite group which the order of 2-Sylow subgroup is 27,the order of the highest order element is 21,the order of the next highest order element is 15,then the following One is established:(1) G/H≌H12·2,H is a nilpotent group of exponent is 7 and of the order is 7c or H is a nilpotent group of exponent dividing 21 and of the order is 3a-3·7c.(2) G/H≌M22,H is a nilpotent group of exponent is 3 and of the order is 3a-2 or H is a nilpotent group of exponent is 5 and of the order is 5b-1 or H is a nilpotent group of exponent is 7 and of the order is 7c～1 or H is a nilpotent group of exponent dividing 15 and of the order is 3a-2·5b-1 or H is a nilpotent group of exponent dividing 21 and of the order is 3a-3·7c-1.(3) G/H≌U3 (4)·2,H is a nilpotent group of exponent is 7 and of the order is 7c or H is a nilpotent group of exponent dividing 21 and of the order is 3a-1·7c.(4) G/H≌G2(3)·2,H is a nilpotent group of exponent is 5 and of the order is 56 or H is a nilpotent group of exponent dividing 15 and of the order is 3a-6·5b.(5) G/H≌L4(3),H is a nilpotent group of exponent is 7 and of the order is 7c or H is a nilpotent group of exponent dividing 21 and of the order is 3a-6·7c.(6) The simple truncation of G/Soc(G) is either all {2,3,5}-groups or all{2,3,7}-groups,and when G/Soc(G) contains two simple truncations,the simple truncation can only be A5.If the simple truncation of G/Soc(G) is {2,3,5}-group,then 7||Soc(G)|;If the simple truncation of G/Soc(G) is {2,3,7}-group,then 5||Soc(G)|.(7) G has only one truncation 1 (?)N (?)K (?)G.where K/N is one of A7,A8,L3(4),U3 (5),A9, J2,A10,U4(3),N is solvable,G/K is {2,3}-group.