Locally finite Artinian p-group implies hypercentral

Statement
Any locally finite Artinian fact about::p-group is hypercentral: the transfinite fact about::upper central series of the group terminates at the whole group.

Locally finite group
A group is termed locally finite if every finitely generated subgroup of the group is finite.

Artinian group
A group is termed Artinian if every nonempty collection of subgroups has a minimal element: a subgroup in that collection that does not contain any other member of that collection.

p-group
A group is termed a p-group for some prime $$p$$ if the order of every element of the group is a power of the prime $$p$$.

Facts used

 * 1) Local finiteness is quotient-closed
 * 2) Artinianness is quotient-closed
 * 3) p-group property is quotient-closed
 * 4) Prime power order implies nilpotent

Reduction
By facts (1), (2), (3), and transfinite induction, it suffices to prove that the center of a nontrivial locally finite Artinian p-group is nontrivial. We do this proof.

Proof of nontrivial center
Given: A prime $$p$$, a nontrivial locally finite Artinian $$p$$-group $$P$$.

To prove: $$Z(P)$$ is nontrivial.

Proof:


 * 1) The centralizer of any finitely generated subgroup of $$P$$ is nontrivial (uses local finiteness): Observe that if the finitely generated subgroup is trivial, then its centralizer is $$P$$, which by assumption is nontrivial. Otherwise, by the local finiteness condition, the finitely generated subgroup is a nontrivial finite $$p$$-group, i.e., a group whose order is a power of $$p$$. By fact (4), this subgroup has a nontrivial center, and thus, its centralizer, which must contain the center, is also nontrivial.
 * 2) The set of centralizers of finitely generated subgroups is closed under finite intersections (uses nothing): Now, let $$S$$ be the set of centralizers of finitely generated subgroups. Note that if $$A$$ and $$B$$ are finitely generated, then so is $$\langle A, B \rangle$$, and $$C_P(\langle A, B \rangle) = C_P(A) \cap C_P(B)$$. Thus, $$S$$ is closed under finite intersections.
 * 3) The set of centralizers of finitely generated subgroups has a member contained in every other member (uses Artinianness): By Artinianness of $$P$$, there exists a finitely generated subgroup $$A$$ such that $$C_P(A)$$ is minimal among elements of $$S$$. For any finitely generated subgroup $$B$$, $$C_P(A) \cap C_P(B) \le C_P(A)$$, and since $$S$$ is closed under finite intersections, we have a member of $$S$$ contained in $$C_P(A)$$. By the minimality of $$C_P(A)$$, we get $$C_P(A) \cap C_P(B) \le C_P(A)$$, forcing $$C_P(A) \le C_P(B)$$ for all finitely generated subgroups $$B$$.
 * 4) The center is nontrivial:  In particular, $$C_P(A)$$ centralizes every cyclic subgroup of $$P$$. Thus, $$C_P(A)$$ centralizes every element of $$P$$. But this forces $$C_P(A) \le Z(P)$$. Since $$C_P(A)$$ is nontrivial, we obtain that $$Z(P)$$ is nontrivial.