A. Solvable Groups.
Before proceeding with the applications we must discuss certain questions in the theory of groups. We shall assume several simple propositions: (a) If N is a normal subgroup of the group G, then the mapping f(x) = XN is a homomorphism of G on the factor group G/N. f is called the natural homomorphism. (b) The image and the inverse image of a normal subgroup under a homomorphism is a normal subgroup, (c) If f is a homomorphism of the group G on G' , then setting N1 = f(N), and defining the mapping g as g( xN ) = f(x) N 1 , we readily see that g is a homomorphism of the factor group G/N on the factor group G '/N' . Indeed, if N is the inverse image of N ' then g is an isomorphism.
We now prove
THEOREM 1. (Zassenhaus). If U and V are subgroups of G, U and v normal subgroups of U and V, respectively, then the following three factor groups are isomorphic:
It is obvious that U r\ v is a normal subgroup of U n V. Let f and from which it follows
Then and be the natural mapping of U on U/u. Call that is isomorphic to H/K. If however, we view f as defined only over then is also isomorphic to H/K.
Thus the first and third of the above factor groups are isomorphic to each other. Similarly, the second and third factor groups are isomorphic.
Corollary 1. If H is a subgroup and N a normal subgroup of the group G, then H/HON is isomorphic to HN/N, a subgroup of G/N.
Proof: Set G = U, N = u, H = V and the identity 1 - v in Theorem 1.
Corollary 2. Under the conditions of Corollary 1, if G/N is .abelian, .so also is H/HfiN.
Let us call a group G solvable if it contains a sequence of sub groups each a normal subgroup of the preceding, and witn G. /Gi abelian.
THEOREM 2. Any subgroup of a solvable group is solvable. For play abelian follows from Corollary 2 above, where let H be a subgroup of G, and call the role of G, N and H.
THEOREM 3. The homomorph of a solvable group is solvable.
where Gj belongs to a and define a sequence exhibiting the solvability of G. Then by (c) there exists a homomorphism mapping
But the homomorphic image of an abelian group is abelian so that the groups G ^ exhibit the solvability of G ' .