Statements in group theory which imply deep results in number theory

The notion of arithmetically equivalent number fields is a good example of a connection between group theory and number theory, see for example:
http://sbseminar.wordpress.com/2007/08/29/zeta-function-relations-and-linearly-equivalent-group-actions/

a couple of specific applications:

Lemma: Let $G$ be a finite $p$-group. Any two subgroups of index $p$ are quasi-conjugated.

Corollary: Two number fields $K$, $L$ of degree $p$ prime are arithmetically equivalent if and only if $[KL:Q] neq p^2$
See "A remark about zeta functions of number fields of prime degree" by R. Perlis.

Also by doing some basic group theory one can prove that any two arithmetically equivalent number fields of degree less than $7$ must be isomorphic.(This is also proven in a paper by Perlis but I don't remember what paper.)

Another result that comes to my mind with this question (totally unrelated to arithmetical equivalence) is that every group of odd order can be realized as a Galois group over Q(odd order theorem plus Shafarevich).

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