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Counterexample that $a\times b\times c = \sum_{(a,b,c)=1} abc$ does not always hold

I was wondering if anyone knows if there is an example of (finite) group $G$ and $g \in G$ where we have $|G| = a\times b\times c = \sum_{(a,b,c)=1} abc$ does not hold for every integer triple $(a,b,c)$.

A:

This is possible. Here’s an example:

$G$ has order $12$ and is generated by three elements $x, y, z$ with orders $3,3,4$.
Then, letting $d=x^{ -1}z$, $e=z^{ -1}y^{ -1}yz$, $f=y^{ -1}yz$, $d^2=e^{2}=f^3=1$, $d^4=e^4=f^4=1$, $d^{ -3}=e^{ -3}=f^{ -3}=1$, $d^{ -4}=e^{ -4}=f^{ -4}=1$, we get $|G|=12=a\times b\times c$, $a=3,b=3,c=4$ and it follows that $d$ and $f$ are not necessarily in the base 3 expansion of $G$.
However, I can’t imagine what an example would be for an infinite group.

A:

One quite simple counterexample has order $12$ and has $3$ in the center (meaning that $a=b=c$)
$$H = \langle a, b, c \mid a^2 = b^2 = c^2 = (ab)^3 = (bc)^3 = 1 \rangle$$
and has \$H = H’ =

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