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Although it is certainly important to study frequently seen group operations like permutations, function composition, word operations, and so on, I find it fascinating to see group structure applied in strange and complicated ways to other disciplines.

Lately MSE has got me looking at the group of functions $f:\mathbb{N}\rightarrow \mathbb{C}$ with $f(1)\not= 0$ under the Dirichlet product $\star$ given by $$(f\star g)(n)=\sum_{ab=n}f(a)g(b).$$ I think this is a great example of an unusual group operation. I don't usually get into this type of arcane number theoretic stuff and never would have thought to look at this type of structure myself.

What are some other examples of obscure group operations I may not have heard of?

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That particular group is the group of units of a commutative ring, fwiw. Lots of other cases similar to this in rings. – Thomas Andrews Dec 30 '12 at 23:06
Right; it's the group of units of a (reduced) incidence algebra: . This is not particularly arcane stuff! More generally, any procedure to construct interesting rings automatically constructs interesting groups. – Qiaochu Yuan Dec 30 '12 at 23:12
up vote 2 down vote accepted

The Butcher group is an infinite-dimensional Lie group arising in numerical analysis and ODEs. It is related to the Connes-Kreimer Hopf algebra, which appears in Connes and Kreimer's approach to understanding renormalization in quantum field theory.

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A well-known but IMO somewhat surprising one is the group law on elliptic curves.

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And this is the only example I know that the hardest part to show, by far and away, is associativity.... – DonAntonio Dec 31 '12 at 2:48
In my extremely limited experience dealing with groups, it seems like if you get the structure from some larger structure then associativity is very easy, but if you're building something more or less from nothing then associativity is very hard. Since most of the groups I have seen are subsets of functions, associativity sort of comes with the package. – Eric Stucky Dec 31 '12 at 17:27

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