Artinian implies co-Hopfian
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., Artinian group) must also satisfy the second group property (i.e., co-Hopfian group)
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Further information: Artinian group
A group is termed Artinian if it satisfies the following equivalent conditions:
- If is a descending chain of subgroups, there is a such that for all .
- Any nonempty collection of subgroups of has a minimal element: a subgroup not containing any other member of that collection.
Further information: co-Hopfian group
A group is termed co-Hopfian if there is no proper subgroup of isomorphic to .
- Slender implies Hopfian: An ascending chain condition on subgroups implies that the group is not isomorphic to any proper quotient.
- Ascending chain condition on normal subgroups implies Hopfian: In fact, an ascending chain condition on normal subgroups implies that the group is not isomorphic to any proper quotient.
We prove the contrapositive here: if a group is not co-Hopfian, it is not Artinian.
Given: A group that is not co-Hopfian.
To prove: is not Artinian.
Proof: Suppose is a subgroup and is an isomorphism (such a subgroup exists because is not co-Hopfian). Define:
We prove by induction that is a proper subgroup of for each . The base case is direct, since .
For the induction, suppose . Since is an isomorphism, it preserves strictness of inclusions, and we thus have:
Thus, we have a strictly descending chain of subgroups of that does not stabilize at any finite stage. Thus, is not Artinian.