Shen's conjecture on groups with given same order type

For any group Gý, ýwe define an equivalence relation ∼ as belowý: ý
∀ gý,ýh∈G g∼h⟺|g|=|h|
ý ýthe set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denote by α(G)ý. ýIn this paperý, ýwe give a partial answer to a conjecture raised by Shený. ýIn factý, ýwe show that if G is a nilpotent groupý, ýthen |π(G)|≤|α(G)|ý, ýwhere π(G) is the set of prime divisors of order of Gý. ýAlso we investigate the groups all of whose proper subgroupsý, ýsay H have |α(H)|≤2ý.