Divisibility-closedness is not finite-join-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., divisibility-closed subgroup) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).
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Statement

It is possible to have a group G and subgroups H,K of G such that both H and K are divisibility-closed subgroups of G but the join of subgroups \langle H, K \rangle is not a divisibility-closed subgroup of G.

Related facts

Proof

Suppose G is the generalized dihedral group corresponding to the additive group of rational numbers. Let H and K both be subgroups of order two generated by different reflections. Then, the following are true:

  • G is divisible by all primes other than 2.
  • H and K are both divisibility-closed subgroups on account of being finite groups.
  • \langle H, K \rangle is isomorphic to the infinite dihedral group. It is not divisible by any primes, and in particular it is not divisibility-closed in G.