Central factor is not finite-intersection-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Intersection-closed subgroup property (?), .
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Statement
An intersection of central factors need not be a central factor.
Proof
Further information: dihedral group:D8
We construct a counterexample as follows. Let where
is the dihedral group of order eight, given by the presentation:
,
and is the cyclic group on two elements, with generator
.
Look at the subgroups and
. We have the following:
-
is a direct factor, and in particular, a central factor.
-
is an automorph of
under the automorphism of
given by
. Thus,
is also a direct factor of
, and hence, a central factor.
- The intersection
is given by
. This is an abelian subgroup, but it is clearly not in the center, hence it cannot be a central factor.
Note that both and
are direct factors, so the proof shows that an intersection of direct factors need not be a central factor. In fact, the same example shows that many related properties are not closed under finite intersections.