%0 Journal Article
%T Shen's conjecture on groups with given same order type
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Jafari Taghvasani, Leyli
%A Zarrin, Mohammad
%D 2017
%\ 03/01/2017
%V 6
%N 1
%P 17-20
%! Shen's conjecture on groups with given same order type
%K Nilpotent groups
%K Same-order type
%K Schmidt group
%R 10.22108/ijgt.2017.10631
%X 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 whose proper subgroups, say $H$ have $|\alpha{(H)}|\leq 2$.
%U https://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf