Finite implies subnormal join property

Verbal statement
Any finite group satisfies the subnormal join property. In other words, a join of finitely many subnormal subgroups of a finite group is again subnormal. Note that since a finite group has only finitely many subnormal subgroups, this also shows that any finite group satisfies the generalized subnormal join property.

Stronger facts

 * Slender implies subnormal join property
 * Ascending chain condition on subnormal subgroups implies subnormal join property
 * Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property
 * Subnormal of finite index implies join-transitively subnormal: The join of any subnormal subgroup of finite index and any subnormal subgroup is subnormal.

Other related facts

 * Nilpotent derived subgroup implies subnormal join property
 * 2-subnormal implies join-transitively subnormal
 * 3-subnormal implies finite-conjugate-join-closed subnormal
 * Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal

Facts used

 * 1) Any finite group is a group satisfying ascending chain condition on subnormal subgroups, i.e., it cannot have an infinite ascending chain of subnormal subgroups.
 * 2) uses::Ascending chain condition on subnormal subgroups implies subnormal join property

Proof
The proof follows directly from facts (1) and (2).