normal subgroup lattice is modular

The fact that the normal subgroupsMathworldPlanetmath of a group G form a latticeMathworldPlanetmath (call it N(G)) is proved here ( The only remaining item is to show that N(G) is modular ( This means, for any normal subgroups H,K,L of G such that LK,


where the meet operationMathworldPlanetmath AB denotes set intersectionMathworldPlanetmath AB, and the join operation AB denotes the subgroupMathworldPlanetmathPlanetmath generated by AB.


First, we show that L(HK)(LH)K. It is easy to see that

  1. 1.

    LK(LH): LK is assumed and LLH follows from the definition of , and

  2. 2.

    HKK(LH): HKK follows from the definition of , and HKHLH.

As a result, L(HK)K(LH)=(LH)K.

Before proving the other inclusion, we shall derive a small lemma concerning LH where L,H are normal subgroups of G:

LH={hL and hH}.

One direction is obvious, so we will just show LH{hL and hH}. Any element of LH can be expressed as a finite productPlanetmathPlanetmath of elements from L or H. This finite product representation can be reduced so that no two adjacent elements belong to the same group. Next, h=(hh-1)h, where hh-1L since L 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 L, 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 inductionMathworldPlanetmath argument) so it has the final form h. ∎

Now back to the main proof. Take any g(LH)K. Then gK and gLH and so g=h for some L and hH by the lemma just shown. Since gK, this means h=-1gLKK. So hHK. We have just expressed g as a product of L and hHK, and so gL(HK). ∎

Title normal subgroup lattice is modular
Canonical name NormalSubgroupLatticeIsModular
Date of creation 2013-03-22 15:50:33
Last modified on 2013-03-22 15:50:33
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Derivation
Classification msc 06C05
Classification msc 20E25