The Hopfian property of n-periodic products of groups

LetH be a subgroup of a groupG. A normal subgroupN _H ofH is said to be inheritably normal if there is a normal subgroup N _G of G such that N _H = N _G ∩ H. It is proved in the paper that a subgroup $N_{G_i }$ of a factor G _i of the n-periodic product Π _i∈I ⁿ G _i with nontrivial factors G _i is an inheritably normal subgroup if and only if $N_{G_i }$ contains the subgroup G _i ⁿ . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π _i∈I ⁿ G _i contains the subgroup G ⁿ . It follows that almost all n-periodic products G = G _{1 *} ⁿ G ₂ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.