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) not satisfying 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.