Group Chains, Subgroup and Factor Group of a Solvable Group

Subgroup and Factor Group of a Solvable Group

Let G be a solvable group with series N, and let H be a subgroup of G. Intersect N_i with H to create a new descending chain of subgroups. Conjugate H∩N_i+1 by anything in H∩N_i, and find a subgroup that is contained in N_i+1 and in H. The restriction to H always produces normal subgroups, hence we have a subnormal series for H.

A homomorphism F takes N_i onto a quotient group Q, with kernel N_i+1. By the correspondence theorem, a subgroup in the domain produces a subgroup in the range, so restrict to H, and F produces a subgroup of Q. If Q is abelian then every subgroup of Q is also abelian. The factor groups of our new, restricted series are all abelian. If G is solvable then H is solvable.

Next consider the image of a solvable group G, under a homomorphism F. Let the image be Q, and let Q_i be the image of N_i under F. The image of a normal subgroup is normal, so each Q_i+1 is normal in Q_i. We have a series for Q.

Remember that the quotient N_i/N_i+1 is abelian. If A and B are cosets of N_i+1 in N_i, AB = BA. Applying F shows that the cosets of F(N_i+1) commute as well, and the factor group Q_i/Q_i+1 is abelian.

In summary, the subgroup or homomorphic image of a solvable group is solvable. As it turns out, the converse is also true.

Let K be normal in G, with factor group H, and assume H and K are solvable. Find a solvable series for H and take the preimage under F. This gives a descending chain of subgroups in G, each normal in the previous. The last subgroup is K.

Next, append the solvable series for K. This takes us all the way down to e. We have a well defined series, but are all the factor groups abelian?

Certainly the series for K meets our criteria, so consider the preimages of H. Let J_i and J_i+1 be the preimages of H_i and H_i+1 respectively. By correspondence under F, J_i/J_i+1 is isomorphic to H_i/H_i+1. Since the latter is abelian, so is the former. Therefore G is solvable.

Finite P Groups are Solvable

Let G be a p group with nontrivial center C. The center is solvable, and the quotient group is a smaller p group, hence solvable. by the above theorem, G is solvable.