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When thinking about groups, I often find that it helps me to think in terms of some "natural examples" which arise outside of group theory. E.g. to my mind general abelian groups are like vector spaces, infinite nonabelian groups are like matrix groups over $\mathbb R$ or $\mathbb C$, and finite abelian groups are products of cyclic groups. The first two of these are familiar to me from linear algebra and the third from elementary number theory.

With finite nonabelian groups I can't think of any such examples, and large swathes of group theory (e.g. the Sylow Theorems) seem only concerned with this case. This is not for lack of groups: I can read about symmetric groups, alternating groups, matrix groups over finite fields, everything discussed at this question, but none of these strike me as "natural" since I don't know of any purpose for them other than as examples and counterexamples in group theory itself. I imagine this has more to do with my own ignorance than with the groups themselves, hence this question.

So what I'm looking for are examples of classes of finite nonabelian groups appearing prominently in some subject outside of group theory, with a description of what role they play in that subject. I would prefer classes of groups to individual groups, and I would prefer relatively accessible subjects over those that are somewhat niche or require a lot of group theory to understand. In particular I'm not interested in the sort of technical, applications-focused answers such as the ones found here.

Thank you!

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  • $\begingroup$ What's wrong with matrix groups over finite fields ? $\endgroup$ – Max Jan 6 '19 at 15:07
  • $\begingroup$ Absolutely nothing, I simply don't know much about their appearance in other areas. I'd be happy to be pointed to where I could learn more about them. $\endgroup$ – namsos Jan 6 '19 at 15:10
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    $\begingroup$ Automorphism groups of finite structures such as finite graphs. $\endgroup$ – Somos Jan 6 '19 at 17:23
  • $\begingroup$ "I don't know of any purpose for them other than as examples and counterexamples in group theory itself." This is getting things backwards. Many examples of nonabelian groups were well known before group theory was fully formalized. I might even argue that one can find examples in Euclid. The group of symmetries of a square is not abelian: any geometer worth their salt knows that reflection across a line that bisects a square does not commute with $90^\circ$ rotation. $\endgroup$ – Lee Mosher Jan 9 '19 at 22:59
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Galois theory is full of finite nonabelian groups. The Galois group of a field extension seems like a very natural thing to study (at least with hindsight), in a way it's a natural generalization of the complex conjugation.

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Symmetry groups are not, as you say, only examples and counterexamples in group theory itself. They have really a lot to do with geometry and physics, among other things. Some arguments are given here:

Applications of group theory to geometry

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Polya theory: The number of distinct $k$-ary bracelets of length $n$ (i.e., bracelets with $n$ beads organized in a regular $n$-gon, where the beads can have one of $k$ colours) is computed by the action of the Dihedral group $D_n$ (consisting of rotations and reflections) on the set of such bracelets. The group $D_n$ is the (nonabelian) semidirect group of the cyclic groups $C_n$ and $C_2$

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