The Hopfian property of n-periodic products of groups

Let H be a subgroup of a group G . A normal subgroup N H of H 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 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 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.