A remark on normal closures in free products of groups
Dali Zangurashvili
TL;DR
The paper investigates normal closures in free products of nontrivial groups and proves that a nontrivial normal subgroup N of a factor sits as a free factor in its normal closure H inside the free product. It uses the Kurosh subgroup theorem and MacLane's uniform Schreier systems to obtain a detailed decomposition of subgroups in free products via i-systems and i-double representatives. The main result shows that H = N * K with a complement K unique up to isomorphism, and that H ∩ G_i = N. This provides canonical free-factor decompositions of normal closures in free products, clarifying the internal structure of subgroups in amalgams and aiding further subgroup analysis in such constructions.
Abstract
It is shown that a nontrivial normal subgroup $N$ of a group $G$ is a free factor of the $N$'s normal closure in the $G$'s free product with arbitrary nontrivial groups.