@article {
author = {Jafari Taghvasani, Leyli and Zarrin, Mohammad},
title = {Shen's conjecture on groups with given same order type},
journal = {International Journal of Group Theory},
volume = {6},
number = {1},
pages = {17-20},
year = {2017},
publisher = {University of Isfahan},
issn = {2251-7650},
eissn = {2251-7669},
doi = {10.22108/ijgt.2017.10631},
abstract = {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 $|\alpha{(H)}|\leq 2$.},
keywords = {Nilpotent groups,Same-order type,Schmidt group},
url = {https://ijgt.ui.ac.ir/article_10631.html},
eprint = {https://ijgt.ui.ac.ir/article_10631_10c392058c06b47129bcb68f68318e72.pdf}
}