Central factor is upper join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about central factor |Get facts that use property satisfaction of central factor | Get facts that use property satisfaction of central factor|Get more facts about upper join-closed subgroup property
Statement with symbols
- Central factor satisfies intermediate subgroup condition
- Central factor satisfies image condition
- Central factor is transitive
- Central factor does not satisfy transfer condition
Using function restriction expressions
This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic
Given: is a group, is a subgroup of , and , are subgroups of all containing . is a central factor of each .
To prove: is a central factor of the join of s. In other words, if is an element in the join of the s, then there exists such that conjugation by agrees with conjugation by on .
Proof: Suppose is an element in the join of the s. Then, there exist and such that .
For each , since is a central factor of , there exists such that conjugation by agrees with conjugation by on . Set . Then, since conjugation is a group action, we see that conjugation by agrees with conjugation by on , completing the proof.