shen's conjecture on groups with given same order type

‎‎for any group $g$‎, ‎we define an equivalence relation $thicksim$ as below‎: ‎[forall g‎, ‎h in g gthicksim h longleftrightarrow |g|=|h|]‎ ‎the set of sizes of equivalence classes with respect to this relation is called the same-order type of $g$ and denote by $alpha{(g)}$‎. ‎in this paper‎, ‎we give a partial answer to a conjecture raised by shen‎. ‎in fact‎, ‎we show that if $g$ is a nilpotent group‎, ‎then $|pi(g)|leq |alpha{(g)}|$‎, ‎where $pi(g)$ is the set of prime divisors of order of $g$‎. ‎also we investigate the groups all of whose proper subgroups‎, ‎say $h$ have $|alpha{(h)}|leq 2$‎.