# Artinian implies periodic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Artinian group) must also satisfy the second subgroup property (i.e., periodic group)
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## Statement

An Artinian group (i.e., a group in which every descending chain of subgroups stabilizes at a finite stage) must be a periodic group: every element in the group has finite order.

## Proof

We prove the contrapositive.

Given: A group $G$ and an element $g \in G$ of infinite order.

To prove: $G$ is not Artinian.

Proof: Consider the descending chain of subgroups:

$\langle g \rangle \ge \langle g^2 \rangle \ge \langle g^4 \rangle \ge \dots$.

Since $g$ has infinite order, this is a strictly descending chain of subgroups that never stabilizes. Thus, $G$ is not Artinian.