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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We prove the contrapositive.
Given: A group and an element of infinite order.
To prove: is not Artinian.
Proof: Consider the descending chain of subgroups:
Since has infinite order, this is a strictly descending chain of subgroups that never stabilizes. Thus, is not Artinian.