Let G be a group, define g h if g , h G have the same order. The set of sizes of the equivalence dir=ltr For instance, if G is torsion free, its type is {1, |G|} . The only group of type {1} are 1 , Z 2. Note that the same-order type {n 1 , n 2 , ... , n r } means n i n j for i j . In 1953 Itô introduced the concept of conjugate types. Let G be a finite group and {n 1 , n 2 , …, n r } the set of integers each of wich is the index of the centralizer of some element of G in G . We may assume that n 1 n 2 … n r =1 . Then the conjugate type vector n 1 , n 2 , … , n r is called the conjugate type vector of G . A group with the conjugate type vector (n 1 , n 2 , ... , n r ) is said to be a group of conjugate type (n 1 , n 2 , ... , n r ) . Obviously , a group of type (1) is only abelian. Itô has proved that finite groups of conjugate type {1, n 1 } and {1, n 1 , n 2 } are nilpotent and solvable , respectively. Afterwards, he determined finite simple groups of conjugate type (1, n 1 , n 2 , n 3 ) . By analogy, we replace the size of the conjugate dir=ltr Let p be the ltr"
