normal subgroups form sublattice of a subgroup lattice
First, we show that is closed under . Suppose and are normal subgroups of . If , then for any , since is normal, and likewise. So , implying that is normal in , or .
To see that is closed under , let be normal subgroups of , and consider an element
where or . If , then
where each or . Therefore, , so is normal in and .
Since is closed under and , is a sublattice of .
Remark. If is finite, it can be shown (Wielandt) that the subnormal subgroups of form a sublattice of .
- 1 H. Wielandt Eine Verallgemeinerung der invarianten Untergruppen, Math. Zeit. 45, pp. 209-244 (1939)
|Title||normal subgroups form sublattice of a subgroup lattice|
|Date of creation||2013-03-22 15:48:24|
|Last modified on||2013-03-22 15:48:24|
|Last modified by||CWoo (3771)|