normal subgroups form sublattice of a subgroup lattice

Consider L(G), the subgroup lattice of a group G. Let N(G) be the subset of L(G), consisting of all normal subgroupsMathworldPlanetmath of G.

First, we show that N(G) is closed under . Suppose H and K are normal subgroups of G. If xHK=HK, then for any gG, gxg-1H since H is normal, and gxg-1K likewise. So gxg-1HK=HK, implying that HK is normal in G, or HKN(G).

To see that N(G) is closed under , let H,K be normal subgroups of G, and consider an element


where xiH or xiK. If gG, then


where each gxig-1H or K. Therefore, gxg-1HK, so HK is normal in G and HKN(G).

Since N(G) is closed under and , N(G) is a sublattice of L(G).

Remark. If G is finite, it can be shown (Wielandt) that the subnormal subgroupsMathworldPlanetmath of G form a sublattice of L(G).


  • 1 H. Wielandt Eine Verallgemeinerung der invarianten Untergruppen, Math. Zeit. 45, pp. 209-244 (1939)
Title normal subgroups form sublattice of a subgroup lattice
Canonical name NormalSubgroupsFormSublatticeOfASubgroupLattice
Date of creation 2013-03-22 15:48:24
Last modified on 2013-03-22 15:48:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Example
Classification msc 20E15