The introduction to group theory that I'm reading requires that the actions of a group are "deterministic"; but the formal definition given makes no mention of this property:
A set G is a group if the following criteria are satisfied.
- There is a binary operation $\cdot$ on $G$.
- That operation is associative...
- There is an identity element $e$...
- Every element in $g\in G$ has an inverse...
Neither do other definitions I'm familiar with (which only mention the operation and the axioms of closure, associativity, identity element, and inverse element).
Where did this property go? Is it really a property of groups or only of certain kinds of groups? Does it follow from other properties or axioms?
I'm guessing that another way to express "deterministic" is that for $a,b,c\in G$, $a=c \land b=c \implies a=b$, but I don't see that in the definition either.