If each Ni is normal in G, our subnormal series is also called a normal series for G. Sometimes the phrase "normal series" is used when the distinction between a normal series and a subnormal series is not important. We just say N is a normal series for G.
The factors of a subnormal series are the factor groups Ni/Ni+1. If the factor groups are all abelian, the series is "solvable". A group is solvable if it has a solvable series. Every abelian group is solvable, for example; just set N0 = G and N1 = e.
Let's look at some nonabelian solvable groups. The dihedral group Dn represents the reflections and rotations of the regular n-gon. The rotations alone form the cyclic subgroup Zn. This is a normal subgroup of index 2. Thus we can write the series Dn, Zn, e, having factor groups Z2 and Zn. Every dihedral group is solvable.
The symmetric groups S2, S3, and S4 are solvable. The latter has the normal series S4, A4, {1234, 2143, 3412, 4321}, and e, with factors Z2, Z3, and Z2Z2.
Larger symmetric groups are not solvable. We will show, later, that every subgroup of a solvable group is solvable, and An, a subgroup of Sn, is simple, and is not solvable.
A refinement of a series inserts a subgroup H between Ni and Ni+1, such that H is normal in Ni, and Ni+1 is normal in H. Of course we may do this many times; the result is still a refinement of the original series. Keep in mind, a refinement of a normal series could become a subnormal series. Just insert a subgroup H that is not normal in G. The quaternion groups provide an example. The series {Q24 Q8 e} is normal, but insert Z4, generated by i, and find the subnormal series {Q24 Q8 Z4 e}. This because [1,i,-1,-i] is not normal in Q24.
A refinement is always possible whenever the factor group Ni/Ni+1 is not simple. Let's illustrate with a solvable group.
Let N be a solvable series for G, and insert the subgroup H to produce a refinement of N. By the correspondence theorems, we can mod out by Ni+1. Now H maps to a subgroup of Ni/Ni+1, which happens to be abelian, so the image of H is automatically normal in the factor group. The image of H is also abelian, hence H/Ni+1 is abelian. Furthermore, the factor Ni/H is the homomorphic image of the abelian group Ni/Ni+1, so this factor is also abelian. Therefore the refinement of a solvable series remains solvable.
Verify the following for the direct product of groups.
The direct product of arbitrarily many parallel subnormal series is subnormal. If one series is longer than the other, just fill in with e. If we are taking an infinite direct product, there must be a bound on the lengths of the individual subnormal series, else the product series will not reach e in a finite number of steps.
The direct product of arbitrarily many parallel normal series is normal. At any level, the direct product of subgroups, normal within their component groups, remains normal in the product.
At any level, The factor group associated with the direct product of our normal series is equal to the direct product of the individual factor groups.
The direct product of abelian groups is abelian.
The direct product of solvable series is solvable.
The direct product of finitely many solvable groups is solvable. An infinite number of groups could cause trouble, if the solvable series associated with the individual groups increase in length without bound.