Shen's conjecture on groups with given same order type

Abstract

‎‎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 $|alpha{(H)}|leq 2$‎.

Keywords

• ‎Nilpotent groups‎
• ‎Same-order type‎
• ‎Schmidt group