# Complete divisibility-closedness is strongly intersection-closed

From Groupprops

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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## Statement

Suppose is a group and are all completely divisibility-closed subgroups of . Then, the intersection of subgroups is also completely divisibility-closed.

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.

## Related facts

- Divisibility-closedness is not finite-intersection-closed
- Powering-invariance is strongly intersection-closed

## Proof

**Given**: A group , completely divisibility-closed subgroups of . A prime number such that is -divisible. An element . An element such that .

**To prove**:

**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 |