Lattice-complemented does not satisfy intermediate subgroup condition
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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It can happen that is a lattice-complemented subgroup of , and , with not a lattice-complemented subgroup in .
Example of the symmetric group
Further information: symmetric group:S4
Let be the symmetric group on a set of size four.
Consider the subgroups:
is a subgroup of order two and is a subgroup of order four containing . Further:
- is a lattice-complemented subgroup in : The symmetric group on the subset is a lattice complement to in .
- is not a lattice-complemented subgroup in : Indeed, is a cyclic group of order four and is a subgroup of order two, and has no lattice complements.