/sci/ - Science & Math

>>12333048
Nah, don't think so.
Set [math]G = Z \oplus Z[/math], [math]Z[/math] being the integers (not using \mathbb because the example is long).
Set [math]M = Z \oplus 2Z[/math] , [math]N= 2Z \oplus Z[/math], [math]H = 3Z \oplus 2Z[/math] and [math]K = 2Z \oplus 3Z[/math]. Then [math]M + N = G[/math], as requested, all subrgoups are normal, because it's all abelian, and [math]M/H = N/K = Z_3[/math] , which is simple.
But [math]M \cap N = 2Z \oplus 2Z[/math] and [math]H \cap K = 6Z \oplus 6Z[/math], so [math](M \cap N)/(H \cap K) = 3Z \oplus 3Z[/math], which isn't simple.