Crosswordese

The Schur–Zassenhaus theorem is a theorem in group theory which states that if $G$ is a finite group, and $N$ is a normal subgroup whose order is coprime to the order of the quotient group $G/N$, then $G$ is a semidirect product of $N$ and $G/N$.

An alternative statement of the theorem is that any normal Hall subgroup of a finite group $G$ has a complement in $G$.

It is clear that if we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group $C_4$ and its normal subgroup $C_2$. Then if $C_4$ were a semidirect product of $C_2$ and $C_4/C_2\cong C_2$ then $C_4$ would have to contain two elements of order 2, but it only contains one.

The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify groups with a certain set of composition factors?" The other part, which is where the composition factors do not have coprime orders, is tackled in extension theory.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534