Summary:
Let N be a normal subgroup of a group G. A subgroup X of G such that G=NX and
NX=1 is called a complement of N in G. In that case, the group G is called a
split extension of N. One splitting theorem states that a group G always splits
over its normal subgroup N. As a result of this, the group G can be analyzed as
the product of its two subgroups, G=NX and each point in G can be written in a
unique way as g=nx where nN and xX.One of the most fundamental splitting
theorem is the Schur theorem which states that if A is an abelian subgroup of
a group G and (A,G/A)=1 then the group G contains subgroups of order G/A and
each two of them are conjugate in G. In other words, the group G is a split
extension over the subgroup A and each two complements of A in G are conjugate.
The proof of this theorem relies on ideas that later on considered as
fundamental in Cohomology Group Theory.In this dissertation we mention this
relation, and give two different proofs of the Schur-Zassenhaus theorem which
generalizes the Schur theorem (Here the subgroup is not necessarily
abelian).Schur-Zassenhaus theorem: Let G be a finite group and N be a normal
subgroup of G. We assume that (N,G/N)=1. Then the group G contains subgroups of
order G/N and each two of them are conjugate in G.It is worth mentioning that
Feit-Thompson theorem has been used ( e.g A group of odd order is solvable) in
order to prove the existence of the complements in the Schur Zassenhaus
theorem.The first proof relies on tools of Group Theory and the second of
Cohomology Group Theory. To be more specific through the interpretation of the
set H(G,N) as a set of group extensions.In section 1, we give a give a group
theoretic proof of the Schur Zassenhaus theorem.In section 2, we analyze the
group H(G,A) and the interpretation of this as a set of extensions E(G,A) of
A in G. Furthermore, if φ:H>G is a group homomorphism and ξ:M>N is a ZG
homomorphism, then we analyze the maps φ:E(G,A)>E(G,A) and ξ:E(G,M)>E(G,
N).Finally, in section 3, we give fundamental results in Cohomology Group
Theory and through those results we give a second proof of the Schur Theorem.
