Difference between revisions of "Artinian implies co-Hopfian"

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(New page: {{group property implication| stronger = Artinian group| weaker = co-Hopfian group}} ==Statement== Any Artinian group (i.e., a group satisfying the descending chain condition on subg...)
 
 
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A group <math>G</math> is termed co-Hopfian if there is no proper subgroup of <math>G</math> isomorphic to <math>G</math>.
 
A group <math>G</math> is termed co-Hopfian if there is no proper subgroup of <math>G</math> isomorphic to <math>G</math>.
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==Related facts==
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===Similar facts===
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* [[Slender implies Hopfian]]: An ascending chain condition on subgroups implies that the group is not isomorphic to any proper quotient.
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* [[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.
  
 
==Proof==
 
==Proof==
  
We prove the contrapositive here.
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We prove the contrapositive here: if a group is not co-Hopfian, it is not Artinian.
  
 
'''Given''': A group <math>G</math> that is not co-Hopfian.
 
'''Given''': A group <math>G</math> that is not co-Hopfian.
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'''To prove''': <math>G</math> is not Artinian.
 
'''To prove''': <math>G</math> is not Artinian.
  
'''Proof''': Suppose <math>H \le G</math> is a subgroup and <math>\alpha:G \to H</math> is an isomorphism. Define:
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'''Proof''': Suppose <math>H \le G</math> is a subgroup and <math>\alpha:G \to H</math> is an isomorphism (such a subgroup exists because <math>G</math> is not co-Hopfian). Define:
  
 
<math>H_0 = G, H_{i+1} = \alpha(H_i)</math>.
 
<math>H_0 = G, H_{i+1} = \alpha(H_i)</math>.

Latest revision as of 12:56, 18 October 2008

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)
View all group property implications | View all group property non-implications
Get more facts about Artinian group|Get more facts about co-Hopfian group

Statement

Any Artinian group (i.e., a group satisfying the descending chain condition on subgroups) is co-Hopfian: it is not isomorphic to any proper subgroup of itself.

Definitions used

Artinian group

Further information: Artinian group

A group G is termed Artinian if it satisfies the following equivalent conditions:

  • If H_0 \ge H_1 \ge H_2 \ge \dots \ge H_n \ge \dots is a descending chain of subgroups, there is a n such that H_n = H_m for all m \ge n.
  • Any nonempty collection of subgroups of G has a minimal element: a subgroup not containing any other member of that collection.

co-Hopfian group

Further information: co-Hopfian group

A group G is termed co-Hopfian if there is no proper subgroup of G isomorphic to G.

Related facts

Similar facts

Proof

We prove the contrapositive here: if a group is not co-Hopfian, it is not Artinian.

Given: A group G that is not co-Hopfian.

To prove: G is not Artinian.

Proof: Suppose H \le G is a subgroup and \alpha:G \to H is an isomorphism (such a subgroup exists because G is not co-Hopfian). Define:

H_0 = G, H_{i+1} = \alpha(H_i).

We prove by induction that H_{i+1} is a proper subgroup of H_i for each i. The base case is direct, since H_1 = \alpha(H_0) = H < H_0 = G.

For the induction, suppose H_i < H_{i-1}. Since \alpha is an isomorphism, it preserves strictness of inclusions, and we thus have:

\alpha(H_i) < \alpha(H_{i-1}) \qquad \implies H_{i+1} < H_i.

Thus, we have a strictly descending chain of subgroups of G that does not stabilize at any finite stage. Thus, G is not Artinian.