Groupprops, The Group Properties Wiki (pre-alpha)
Take a short survey about Math Resources on the Internet.
Artinian implies periodic
From Groupprops
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)
View all subgroup property implications | View all subgroup property non-implications |Get help on looking up subgroup property implications/non-implications
Get more facts about Artinian group| Get more facts about periodic group
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 
of infinite order.
To prove: G is not Artinian.
Proof: Consider the descending chain of subgroups:
.
Since g has infinite order, this is a strictly descending chain of subgroups that never stabilizes. Thus, G is not Artinian.
