# Permutably complemented is not finite-intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., permutably complemented subgroup)notsatisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).

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## Statement

It is possible to have a group and two permutably complemented subgroups such that is not permutably complemented in .

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Suppose is the dihedral group of order eight:

.

Consider the two subgroups:

.

Then:

.

We have:

- is a permutably complemented subgroup of : The subgroup is a permutable complement to in .
- is a permutably complemented subgroup of : The subgroup is a permutable complement to in .
- is
*not*permutably complemented in : This can be seen by direct inspection, but also follows from the more general fact that in a nilpotent group any nontrivial normal subgroup intersects the center nontrivially. Here, is the center, and if it has a permutable complement, that subgroup must be a nontrivial normal subgroup, leading to a contradiction.