Permutable implies modular

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., permutable subgroup) must also satisfy the second subgroup property (i.e., modular subgroup)
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Statement

Any permutable subgroup of a group is modular.

Definitions used

Permutable subgroup

Further information: Permutable subgroup

A subgroup H of a group G is termed permutable if HK = KH = \langle H, K \rangle for every subgroup K \le G.

Modular subgroup

Further information: Modular subgroup

Facts used

  1. Modular property of groups: This states that if H,K,L are subgroups of G such that H \le L, then:

H(K \cap L) = HK \cap L.

Proof

Given: A subgroup H of a group G such that HK = KH = \langle H, K \rangle for all subgroups K \le G.

To prove: For any subgroups K, L of G such that H \le L, we have:

\langle H, K \cap L \rangle = \langle H, K \rangle \cap L.

Proof: Since H is permutable, we have:

\langle H, K \cap L \rangle = H(K \cap L).

and:

\langle H, K \rangle = HK, so \langle H, K \rangle \cap L = HK \cap L.

Applying fact (1) now yields the result.