# Difference between revisions of "Artinian implies co-Hopfian"

(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>. | ||

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | ===Similar facts=== | ||

+ | |||

+ | * [[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. | ||

==Proof== | ==Proof== | ||

− | We prove the contrapositive here. | + | 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: | + | '''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

## Contents

## 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 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.

### co-Hopfian group

`Further information: co-Hopfian group`

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

## Related facts

### Similar facts

- 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.

## Proof

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.