normal subgroup lattice is modular
The fact that the normal subgroups of a group form a lattice (call it ) is proved here (http://planetmath.org/NormalSubgroupsFormSublatticeOfASubgroupLattice). The only remaining item is to show that is modular (http://planetmath.org/ModularLattice). This means, for any normal subgroups of such that ,
First, we show that . It is easy to see that
: is assumed and follows from the definition of , and
: follows from the definition of , and .
As a result, .
Before proving the other inclusion, we shall derive a small lemma concerning where are normal subgroups of :
One direction is obvious, so we will just show . Any element of can be expressed as a finite product of elements from or . This finite product representation can be reduced so that no two adjacent elements belong to the same group. Next, , where since is normal, showing that an on the right of the product can be “filtered” to the left of the product (of course, this “filtering” changes to another element of , but it is the form of the product, not the elements in the product, that we are interested in). This implies that the finite product representation can further be reduced (by an induction argument) so it has the final form . ∎
Now back to the main proof. Take any . Then and and so for some and by the lemma just shown. Since , this means . So . We have just expressed as a product of and , and so . ∎
|Title||normal subgroup lattice is modular|
|Date of creation||2013-03-22 15:50:33|
|Last modified on||2013-03-22 15:50:33|
|Last modified by||CWoo (3771)|