The Hopfian Property of n-Periodic Products of Groups

摘要

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 N_(Gi) of a factor G_i of the n-periodic product ∏_(i∈I)~n G_i with nontrivial factors G_i is an inheritably normal subgroup if and only if N_(Gi) 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 = ∏_(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.