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Context: I'm re-studying basic group theory and looking for "real-world" examples/puzzles that can be translated into abstract group theoretic statements. By real-world I mean not something necessarily useful but rather what constitutes an interesting problem, e.g. counting the number of non-isomorphic colorings of a square (where the Burnside���s Counting Theorem can be used). In my case, I would like to provide a motivation for the following statement:

Let $C(a)$ be the centralizer subgroup of an element $a \in G$. Then $gC(a)g^{-1} = C(gag^{-1})$.

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  • $\begingroup$ Doesn't this call upfront for a real-world example of the centralizer itself? $\endgroup$
    – Kan't
    Commented Feb 7 at 21:04

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Consider $g$ as providing a coordinate change (matrices) or re-labelling of points (permutations): what commutes with the relabelled $a$ is exactly a relabelling of what commutes with $a$.

Basically this is (using the action of a group on its elements by conjugation) a special case of the statement that the stabilizer of the image of $a$ (under $g$), i.e. $C(a^g)$, are exactly the $g$-conjugate of the stabilizer of $a$: Map the image $a^g$ back to the original point $a$, centralizer there, and map back to $a^g$.

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  • $\begingroup$ Thanks but this just restates the statement in more words but doesn't provide more insight/motivation. Isn't there a more concrete problem, e.g. in the vein of the mentioned coloring counting problem? $\endgroup$
    – David Kubecka
    Commented Feb 7 at 16:37

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