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What are some non-trivial examples of finite subgroups of infinite groups? An example is simpler to understand than another if the background required to understand the example is less than the other.

I discount the answers here and here because they answer the question: "which infinite groups have all infinite subgroups, or all finite subgroups?". (also the answers aren't particularly simple)

This question is dropping the all, and is dropping the request for infinite subgroups. However, this question is not dropping the triviality condition, as the simplest example I can think of is the trivial example: the sub-group containing only the identity element.

bzm3r
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  • What are you non-trivial examples ? There might be too many examples. Just take a finite group $\Gamma_1$ and any other group $\Gamma_2$, then the free product $\Gamma_1*\Gamma_2$ contains $\Gamma_1$ as a subgroup. – M. Dus Sep 30 '18 at 16:49

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The matrix group $M_2(\Bbb{R})$ is infinite, and has a finite cyclic subgroup $C_2$ of diagonal matrices $id,-id$.

Dietrich Burde
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The infinite group $SO(3)$ of all 3d rotations has many finite subgroups, including for example the symmetry groups of solids (e.g., a dodecahedron $\implies A_5$, or a regular prism $\implies D_{2n}$) in $\Bbb R^3$.

Hagen von Eitzen
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