An Implication of the Kurosh–MacLane Subgroup Theorem for Free Products

While MacLane’s version of the Kurosh subgroup theorem for free products has been well-known, one of its implications seems to have escaped the attention of the mathematical community. The purpose of this note is to fill this gap by providing this implication. It asserts that a nontrivial normal subgroup of a group is a free factor of this subgroup’s normal closure in the free product of the group with arbitrary nontrivial groups.