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For any group G, we dene an equivalence relation s as below: 8 g;h 2 G g s h () jgj = jhj the set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denote by (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 j (G)j j (G)j, where (G) is the set of prime divisors of order of G. Also we investigate the groups all of whose proper subgroups, say H have j (H)j 2.

Shen's conjecture on groups with given same order type