  For any group $G$, we define an equivalence relation $\thicksim$ as below:
$$\forall \ g, h \in G \ \ g\thicksim h \Longleftrightarrow |g|=|h|$$ 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
whoseproper subgroups, say $H$ have $|\alpha{(H)}|\leq 2$.
