Law of composition in a group Most textbooks on group theory define the law of composition, say $\star$, as a function $\star: S \times S \to S$. The next part of the definition is sending $(a,b) \mapsto a \star b$. This confuses me somewhat. $\star$ is the function, but we're in effect treating it  as thought it were an operation and writing it between $a$ and $b$ instead of $\star(a,b)$. I'm ok with the abuse of the notation, but the definition is circular, defining $\star$ in terms of itself. One definition I say defines an operation $m: S \times S \to S$ by $(a,b) \mapsto a \cdot b$, but then it doesn't define the symbol $\cdot$, which is really the function/operation.
Is this all just an abuse of notation?
As long as your map $\star$ (or $\cdot$, or etc.) has no property to fulfill, feel free to denote the image of a pair $(a,b)$ of the domain by any between the two equivalent symbols $\star(a,b)$ and $a\star b$. But, as commented by @CharlesHudgins, as soon as you have to deal with properties for your map to fulfill (e.g. associativity), infix notation brings undoubtable practical benefits.
As long as your map $\star$ (or $\cdot$, or etc.) has no property to fulfill, feel free to denote the image of a pair $(a,b)$ of the domain by any between the two equivalent symbols $\star(a,b)$ and $a\star b$. But, as commented by @CharlesHudgins, as soon as you have to deal with properties for your map to fulfill (e.g. associativity), infix notation brings undoubtable practical benefits.