Direct factor is not upper join-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
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Statement

We can have a subgroup H of a group G, and intermediate subgroups K_1 and K_2 such that H is a direct factor of K_1 as well as a direct factor of K_2, but H is not a direct factor of the join of subgroups \langle K_1, K_2.

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider the dihedral group of order eight:

G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle.

Let:

H = \langle a^2 \rangle, K_1 = \langle a^2, x \rangle, K_2 = \langle a^2, ax \rangle.

Then: