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) a (c) H. It is proved in the paper that a subgroup of a factor G (i) of the n-periodic product I (iaI) (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 a parts per thousand yen 665 every nontrivial normal subgroup in a given n-periodic product G = I (iaI) (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.