Lattice-complemented does not satisfy intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., lattice-complemented subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It can happen that H is a lattice-complemented subgroup of G, and H \le K \le G, with H not a lattice-complemented subgroup in K.

Proof

Example of the symmetric group

Further information: symmetric group:S4

Let G be the symmetric group on a set \{ 1,2,3,4 \} of size four.

Consider the subgroups:

H = \langle (1,3)(2,4) \rangle, \qquad K = \langle (1,2,3,4) \rangle.

H is a subgroup of order two and K is a subgroup of order four containing H. Further:

  • H is a lattice-complemented subgroup in G: The symmetric group on the subset \{ 1,2,3 \} is a lattice complement to H in G.
  • H is not a lattice-complemented subgroup in K: Indeed, K is a cyclic group of order four and H is a subgroup of order two, and has no lattice complements.