University of IsfahanInternational Journal of Group Theory2251-76506120170301Shen's conjecture on groups with given same order type17201063110.22108/ijgt.2017.10631ENLeyliJafari TaghvasaniDepartment of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, IranMohammadZarrinUniversity of KurdistanJournal Article20150609For 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 whose proper subgroups, say $H$ have $|\alpha{(H)}|\leq 2$.https://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf