The angle brackets notation in $A=\mathbb{Z}^3/\langle f_1,f_2,f_3 \rangle$.

Question

I am being asked to consider a group $A$, where $A=\mathbb{Z}^3/\langle f_1,f_2,f_3 \rangle$ where $f_1= (4, 0, −14)$, $f_2 = (24, 2, 60)$, and $f_3 = (−5, 0, 20)$.

I don't know what the angle bracket notation means in this context, and as such, I can't even begin to work out what the elements of $A$ will be.

Any help would be much appreciated!

Answer 1

Here $H:=\langle f_1, f_2, f_3\rangle$ is the subgroup of $\Bbb Z^3$ of all elements generated by the $f_i$. That is to say that $H$ contains all elements that are a finite product of both the $f_i$ and/or their inverses, and no other elements.

In this context, it means that one "kills"${}^\dagger$ the $f_i$ by taking the quotient, meaning that one considers all elements of $H$ as equivalent to the identity in $A$.

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$\dagger$: Yes, that's a technical term!