Basic Group Theory

Question

Teaching myself basic group theory as part of a physics self study program and was wondering why we don’t need to axiomatize the concept that if two elements are equal, applying the operator to each results in equal elements.

$x=y$ implies $a•x=a•y$

Answer 1

Group theory is studied in the framework of set theory and, given two sets $A$ and $B$ and a function $f\colon A\longrightarrow B$, it is true that, if $a,b\in A$ and $a=b$, then $f(a)=f(b)$.