FZ implies generalized subnormal join property

Statement
Suppose $$G$$ is a FZ-group (i.e., its center has finite index). Then, $$G$$ is also a group satisfying generalized subnormal join property: an arbitrary join of subnormal subgroups of $$G$$ is subnormal.

Facts used

 * 1) uses::Subnormality satisfies image condition
 * 2) uses::Finite implies generalized subnormal join property
 * 3) Joins commute with images, i.e., the image of a join of subgroups is the join of their individual images.
 * 4) uses::Subnormality satisfies inverse image condition
 * 5) uses::Central factor implies normal

Proof
Given: A FZ-group $$G$$, a collection of subgroups $$\{ H_i \}_{i \in I}$$, all subnormal in $$G$$. $$H = \langle H_i \rangle_{i \in I}$$ is the join of the subgroups.

To prove: $$H$$ is subnormal in $$G$$.

Proof: Let $$Z(G)$$ denote the center of $$G$$.