Cocentrality is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., cocentral subgroup) satisfying a subgroup metaproperty (i.e., transitive 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 cocentral subgroup |Get facts that use property satisfaction of cocentral subgroup | Get facts that use property satisfaction of cocentral subgroup|Get more facts about transitive subgroup property
Statement with symbols
If is a cocentral subgroup of , and is a cocentral subgroup of , then is a cocentral subgroup of .
Further information: Cocentral subgroup
A subgroup of a group is termed cocentral in if , where is the center of .
Given: A group , a cocentral subgroup of , a cocentral subgroup of . In other words, , and .
To prove: is cocentral in , i.e., .
Proof: Consider . Clearly, , and , because centralizes . Thus, , so . Thus, . Thus, , so , and thus . Thus, <mathG = HZ(G)</math>.