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Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?

Here are two examples I thought of:

The existence of Golod-Shafarevich towers of Hilbert class fields follows from an inequality on the dimensions of the first two cohomology groups of the ground field.

Iwasawa's theorem on the size of the $p$ part of the class groups in $\mathbb{Z}_p$-extensions follows from studying the structure of $\mathbb{Z}_p[\![T]\!]$-modules.

Can you name some others?

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I'm not sure that your second example is a straight-forward result coming from group theory, I'd rather think that it is more a result from commutative algebra. Although you could argue that it is in fact a pro-p group representation theoretic result. –  Guillermo Mantilla Dec 3 '09 at 23:20

6 Answers 6

The fact (from class field theory) that ideals become principal in the Hilbert class field follows from the fact that the Verlagerung $V:G^{\text{ab}}\rightarrow H^{\text{ab}}$ is zero if $G$ is any finite group and $H$ is its commutator subgroup.

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Brauer's theorem implies meromorphic continuation of Artin L-functions (indeed, I believe that was Brauer's motivation).

http://en.wikipedia.org/wiki/Brauer's_theorem_on_induced_characters

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I'm sure you omitted this just because it's too classic: big part of group theory was invented to prove that most roots of unity cannot be constructed by quadratic (or cyclic) extensions.

It's still the best, most direct connection between [nt.number-theory] and [gr.group-theory] I know of.

For a more "advanced" version of this, do computations of group cohomology count?

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Even more classic: no-one seems to have mentioned the insolubility of the quintic by radicals follows from the simplicity of the group A_5. Whether that is straightforward depends on what you already know. –  Simon Wadsley Dec 4 '09 at 8:54
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This is exactly what I am saying (I guess I disguised this too well): "big part" = "solvable groups", "cyclic extensions" = solvable by radicals. –  Ilya Nikokoshev Dec 4 '09 at 19:52
    
I don't get it. Roots of unity are in a cyclic extension. Do you mean roots of integer polynomials? –  Dror Speiser Oct 28 '10 at 20:48

I'd say that classification of subgroups of GL$_2(F_p)$ plays a big part in Serre's result about the almost surjectivity of $\ell$-adic Galois representations of CM Elliptic curves.

Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. (French) Invent. Math. 15 (1972), no. 4, 259-331.

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The fundamental theorem of arithmetic (uniqueness of factorization of integers into primes) is an immediate consequence of the Jordan-Holder theorem on uniqueness of composition factors of finite groups.

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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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