I am taking a stab at group theory and in some of the questions I am working on they don't explicitly state whether a group is additive or multiplicative. $\mathbb{Z}_4$ is additive (and that makes sense because integers) but how do I for example determine the operation to use on $\mathbb{R}$? It seems like it could be either one (it satisfies the axioms for both) and it's not clear as to what the question wants. Is there a rule as to what I should do when?
Thanks a lot!
A group $G$ is a group, and as such is endowed with a single binary operation $$\star: \quad G\times G\to G, \qquad(x,y)\mapsto x\star y$$ satisfying certain axioms. If no optional properties of $\star$ are assumed the operation $\star$ is called multiplication in $G$. If in the case at hand the operation $\star$ is commutative, as is the addition of integers, we are inclined to call the operation $\star$ addition, and write $+$ instead of $\star$. This notational move becomes a necessity if on $G$ additional binary operations, like $*$, are installed which interact in a particular way with $+$.