The Main Group Columns Only Supersolvable Group Harmony Groups Of The Column Only Solvable

In the frist part of the article, we give the classification of the supersolvable groups with unique principal series and the classification of the solvable groups with unique composition series. And we give an example to make the difference between a group with unique principal series and a group with unique composition series.In the second part, we give a new approach to prove the number of sylow subgroups of S_n and A_n. In the S_p², we give upbound of the union of two sylow subgroups, we characterise the structure of the center of Sylow p-subgroup of S_n.In the third part, suppose in a group that two elements of finite order are commutative with each other, we give examples to show that the order of their product does not equal to the 1.c.m of their orders.And we get an if-and-only-if condition for two finite elements with the order of their product equals to the l.c.m of their orders. We prove that, for two comunicated p—elemenets a, b, if the order of ab does not equal to the 1.c.m of the order of a and the order of 6, then o(a) and 0(6) must be the same.