Central factor is not finite-intersection-closed

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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 G = D \times C where D is the dihedral group of order eight, given by the presentation:

D = \langle a,x \mid a^4 = x^2 = e, axa^{-1} = x^{-1} \rangle,

and C is the cyclic group on two elements, with generator y.

Look at the subgroups H = \langle x,a \rangle and K = \langle xy,a \rangle. We have the following:

  • H = D \times 1 is a direct factor, and in particular, a central factor.
  • K is an automorph of H under the automorphism of G given by a \mapsto a, x \mapsto xy, y \mapsto y. Thus, K is also a direct factor of G, and hence, a central factor.
  • The intersection H \cap K is given by \langle a \rangle. This is an abelian subgroup, but it is clearly not in the center, hence it cannot be a central factor.

Note that both H and K 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.