Let $G$ be the free abelian group generated by 3 generators, $a$, $b$, $c$.

Technically it should be written as $\langle a,b,c\mid ab=ba, ac=ca, bc=cb\rangle$.

Is there a shorter version of the above notation (that specifies the generators)? That is, we wish to specify that the generators are $a,b,c$.

In particular, is it ever acceptable to write just $\langle a,b,c\rangle$ if the context is clear?

Thanks.

• This depends on taste. For me it is not clear, I would prefer the first one. – Dietrich Burde Mar 20 '18 at 8:48
• An alternative is $\langle a,b,c\rangle_{ab}$. – Tobias Kildetoft Mar 20 '18 at 8:49
• I don't believe that there is a standard notation for this. If I needed to use this a lot, I would write something like ${\rm FAb}\langle a,b,c \rangle$. – Derek Holt Mar 20 '18 at 8:50
• @TobiasKildetoft Thanks. Is there any example where people actually use this notation? – yoyostein Mar 20 '18 at 8:53
• Free abelian groups are isomorphic to $\bigoplus\mathbb{Z}$. Isn't that simplier notation? If you wish to specify $A=\{x,y,z\}$ then $\mathbb{Z}^A$ is an option as well (which is clear when $A$ is finite). I've also seen $\mathbb{Z}^{(A)}$ notation (e.g. wikipedia). – freakish Mar 20 '18 at 9:14