Shen's conjecture on groups with given same order type

‎‎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 ۲‎.

کلیدواژه ها:

‎Nilpotent groups‎ ، ‎Same-order type‎ ، ‎Schmidt group