Central factor is not finite-intersection-closed: Difference between revisions

Latest revision as of 23:24, 15 February 2009

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:

$D = ⟨ a, x ∣ a^{4} = x^{2} = e, a x a^{- 1} = x^{- 1} ⟩$ ,

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

Look at the subgroups $H = ⟨ x, a ⟩$ and $K = ⟨ x y, a ⟩$ . 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 $a \mapsto a, x \mapsto x y, y \mapsto y$ . Thus, $K$ is also a direct factor of $G$ , and hence, a central factor.
The intersection $H \cap K$ is given by $⟨ a ⟩$ . 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.

@@ Line 2: / Line 2: @@
 property = central factor|
 metaproperty = finite-intersection-closed subgroup property|
-corollary1 = intersection-closed subgroup property}}
+first corollary = intersection-closed subgroup property}}
 ==Statement==
@@ Line 10: / Line 10: @@
 ==Proof==
-We construct a counterexample as follows. Let <math>G = D \times C</math> where <math>D</math> is the dihedral group of order 8, and <math>C</math> is the cyclic group on 2 elements. Let <math>a</math> be an element of order 4 in <math>D</math>, and <math>x</math> a reflection in <math>D</math>. Further, let <math>y</math> be the generator of <math>C</math>.
+{{further|[[Particular example::dihedral group:D8]]}}
-Look at the subgroups <math>H = <x,a></math> and <math>K = <xy,a></math>. Both these subgroups are actually automorphs of each other, and moreover, they both have the same centralizer: the group <math><a^2,y></math>. However, the group <math>H \cap K</math>, which is just <math>a</math>, is not a central factor.
+We construct a counterexample as follows. Let <math>G = D \times C</math> where <math>D</math> is the [[dihedral group:D8|dihedral group of order eight]], given by the presentation:
-Note that both <math>H</math> and <math>K</math> are direct factors, so the proof shows that an intersection of direct factors need not be a central factor.
+<math>D = \langle a,x \mid a^4 = x^2 = e, axa^{-1} = x^{-1} \rangle</math>,
+and <math>C</math> is the [[cyclic group:Z2|cyclic group on two elements]], with generator <math>y</math>.
+Look at the subgroups <math>H = \langle x,a \rangle</math> and <math>K = \langle xy,a \rangle</math>. We have the following:
+* <math>H = D \times 1</math> is a direct factor, and in particular, a central factor.
+* <math>K</math> is an automorph of <math>H</math> under the automorphism of <math>G</math> given by <math>a \mapsto a, x \mapsto xy, y \mapsto y</math>. Thus, <math>K</math> is also a direct factor of <math>G</math>, and hence, a central factor.
+* The intersection <math>H \cap K</math> is given by <math>\langle a \rangle</math>. This is an abelian subgroup, but it is clearly not in the center, hence it cannot be a central factor.
+Note that both <math>H</math> and <math>K</math> 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.