FINITE p-GROUPS DETERMINED BY AN INEQUALITY OF THE ORDER OF ANY TWO-ELEMENTS GENERATED SUBGROUPS

Abstract

In this note, we study a class of finite p-group determined by an inequality of the order of any two-elements generated subgroups.  If |\(\langle\)x, y\(\rangle\)| ≤ p²max{|x|,|y|} for all x, y ∈ G, then G is called a M_i-group.  If |\(\langle\)x, y\(\rangle\)| ≤ p²|x| for all x, y ∈ G, then G is called a P_i-group.  Such groups relate to a problem posed by Berkovich and Janko (Groups of prime order, Walter de Gruyter, Berlin, vol. 1, 2008) (Problem 461 and 237).  In this paper, we mainly get the nilpotent class of P_i(or M_i)-groups, the exponent of derived subgroup of Pi(or M_i)-groups and G^{\(p^{2}\)} ≤ Z(G).