Central factor-extensible automorphism: Difference between revisions
(New page: {{wikilocal}} {{variation of|extensible automorphism}} ==Definition== ===Symbol-free definition=== An automorphism of a group is termed '''central factor-extensible''' if, for e...) |
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An [[automorphism]] <math>\sigma</math> of a [[group]] <math>G</math> is termed '''central factor-extensible''' if, for every embedding of <math>G</math> as a [[central factor]] in a group <math>H</math>, there exists an automorphism <math>\varphi</math> of <math>H</math> whose restriction to <math>G</math> is <math>\sigma</math>. | An [[automorphism]] <math>\sigma</math> of a [[group]] <math>G</math> is termed '''central factor-extensible''' if, for every embedding of <math>G</math> as a [[central factor]] in a group <math>H</math>, there exists an automorphism <math>\varphi</math> of <math>H</math> whose restriction to <math>G</math> is <math>\sigma</math>. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Weaker than::Extensible automorphism]] | |||
* [[Weaker than::Normal-extensible automorphism]] | |||
* [[Weaker than::Center-fixing automorphism]]: {{proofat|[[Center-fixing implies central factor-extensible]]}} | |||
==Facts== | |||
In a [[centerless group]], ''every'' automorphism is central factor-extensible. This is because any central factor that is centerless as a group must be a direct factor. | |||
{{further|[[Centerless implies every automorphism is central factor-extensible]]}} | |||
Latest revision as of 21:30, 22 May 2009
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This is a variation of extensible automorphism|Find other variations of extensible automorphism |
Definition
Symbol-free definition
An automorphism of a group is termed central factor-extensible if, for every embedding of the group as a central factor of a group, the automorphism can be extended to an automorphism of the bigger group.
Definition with symbols
An automorphism of a group is termed central factor-extensible if, for every embedding of as a central factor in a group , there exists an automorphism of whose restriction to is .
Relation with other properties
Stronger properties
- Extensible automorphism
- Normal-extensible automorphism
- Center-fixing automorphism: For full proof, refer: Center-fixing implies central factor-extensible
Facts
In a centerless group, every automorphism is central factor-extensible. This is because any central factor that is centerless as a group must be a direct factor.
Further information: Centerless implies every automorphism is central factor-extensible