Complete divisibility-closedness is strongly intersection-closed

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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 G is a group and H_i, i \in I are all completely divisibility-closed subgroups of G. Then, the intersection of subgroups H = \bigcap_{i \in I} H_i is also completely divisibility-closed.

Here, a subgroup is completely divisibility-closed if for any prime number p such that every element of the group has a p^{th} root in the group, all p^{th} roots of any element in the subgroup are in the subgroup.

Related facts

Proof

Given: A group G, completely divisibility-closed subgroups H_i, i \in I of G. A prime number p such that G is p-divisible. An element g \in H = \bigcap_{i \in I} H_i. An element x \in G such that x^p = g.

To prove: x \in H

Proof: It suffices to demonstrate the last sentence, because the existence of p^{th} roots in G is guaranteed by G being p-divisible.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 g \in H_i for each i \in I. H = \bigcap_{i \in I} H_i, g \in H. direct from given
2 x \in H_i for each i \in I. x^p = g, H_i is completely divisibility-closed. Step (1) Step-given direct.
3 x \in H. H = \bigcap_{i \in I} H_i Step (2) Step-given direct