A proof for Abelian groups are solvable

Question

I have seen some proofs for "abelian groups are solvable". Mostly they use induction. However, I came up with my own proof that I have not seen it elsewhere (it might already exist, but I am not aware of it). So I wanted to share my thoughts and see if the proof makes sense:

Let $G$ be an abelian group. Then consider

$$\{1\} \lhd G$$

$\{1\}$ is obviously normal in $G$ and $G/\{1\}$ is isomorphic to $G$, which is abelian.

Does this not prove $G$ is solvable? If so, does this not prove it in general, regardless of the size (i.e. finite vs infinite)?

Answer 1

To answer the question: yes, your proof is fine.

However, the comments uncovered some issues with proofwiki which are worth going into. There are 3 relevant definitions.

1. A group $G$ is solvable if there exists a chain of subgroups $\{1\}=G_1\leq G_2\leq\ldots \leq G_n=G$ such that $G_i\lhd G_{i+1}$ and $G_{i+1}/G_i$ is abelian.

2. A group $G$ is polycyclic if there exists a chain of subgroups $\{1\}=G_1\leq G_2\leq\ldots \leq G_n=G$ such that $G_i\lhd G_{i+1}$ and $G_{i+1}/G_i$ is cyclic.

3. A group $G$ is supersolvable if there exists a chain of subgroups $\{1\}=G_1\leq G_2\leq\ldots \leq G_n=G$ such that $G_i\lhd G$ and $G_{i+1}/G_i$ is cyclic.

The differences are that in (2) and (3) we require the quotients to be cyclic rather than just abelian, and in (3) we require the $G_i$ to be normal in $G$ (so we have a "normal series" rather than just a "subnormal series").

Clearly we have supersolvable$\subset$polycyclic$\subset$solvable. For finite groups, one can prove that polycyclic$=$solvable, which is presumably what proofwiki is aiming for. The group $A_4$ is easily seen to be solvable (as it's derived subgroup is the Klein $4$-group, which is abelian), but is not supersolvable as no non-trivial cyclic subgroup is normal.

Proofwiki's definition of "solvable" actually corresponds to supersolvable, and therefore is incorrect as these two classes of groups are different, even if we only consider finite groups. (To be precise: their definition states $G$ "has a composition series in which each factor is a cyclic group", while they define a composition series to be a specific kind of normal, rather than subnormal, series.)