Complete divisibility-closedness is strongly intersection-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., completely divisibility-closed subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)
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Here, a subgroup is completely divisibility-closed if for any prime number such that every element of the group has a root in the group, all roots of any element in the subgroup are in the subgroup.
- Divisibility-closedness is not finite-intersection-closed
- Powering-invariance is strongly intersection-closed
Given: A group , completely divisibility-closed subgroups of . A prime number such that is -divisible. An element . An element such that .
Proof: It suffices to demonstrate the last sentence, because the existence of roots in is guaranteed by being -divisible.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||for each .||, .||direct from given|
|2||for each .||, is completely divisibility-closed.||Step (1)||Step-given direct.|
|3||.||Step (2)||Step-given direct|