A solvable group seems to be variously defined as one with a composition series where all the composition factors are Abelian, or as one with a subnormal series where all the quotients are Abelian. Unlike with the first definition, this definition does not explicitly seem to require that the quotients be simple. Are they nonetheless equivalent?
Similarly, when looking at e.g. proofs that finite $p$-groups are solvable, I have seen the inductive argument which constructs a subnormal series with all the quotients Abelian but they never seem to show that the quotients are also simple. Why? Can one give a proof which does show that they are simple, and therefore complies with the composition series definition of solvable?
Thanks in advance for assistance.
Best Answer
The definition involving composition series only works for finite groups, so it is just plain wrong as a definition of solvable groups. The group $({\mathbb Z},+)$ is abelian and solvable but has no composition series.
There are two equivalent definitions. Let's call a series $1=N_0 \le N_1 \le N_2 \le \cdots \le N_k=G$ normal if all $N_i \unlhd G$ and subnormal if each $N_{i-1} \unlhd N_i$.
Then a group is solvable if and only if it has a normal series with abelian factors. But this is equivalent to the condition that it has a subnormal series with abelian factors.
To see that, note that if it has a subnormal series with abelian factors, then the derived series of $G$ is a normal series with abelian factors.