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I am trying to understand the proof of:

If $E/F$ is a Galois extension with abelian Galois group, then $E$ is a tower of quadratic extension iff $[E:F]$ is a power of $2$.

It is the proof above. But I cannot understand why we must use quotient here. I think even the subgroup of order $2$ is not normal, we still have the $[E^{<\sigma>}:F]=2^{j-1}$. So why do we need the condition that $G$ is abelian and we need to use the quotient?

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    $\begingroup$ All subgroups of an abelian group are normal. $\endgroup$
    – Alex Vong
    May 23, 2017 at 7:29
  • $\begingroup$ And $E^{\langle\sigma\rangle}$ is Galois over $F$ if and only if $\langle\sigma\rangle\unlhd G$. Furthermore, that is needed to identify $Gal(E^{\langle\sigma\rangle}/F)$ as the quotient group. The quotient group is then automatically abelian, and the induction hypothesis can be applied. $\endgroup$
    – Jyrki Lahtonen
    May 23, 2017 at 7:44

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