Multiplying both sides of an equation in proofs

Question

I'm learning the basics of group theory, and must justify every step of a proof by referring to the basic axioms/theorems. Which axioms/theorems justify multiplying or adding an element of a group to both sides of an equation?

Answer 1

Group operation is first and foremost a map, say $\star\colon G\times G\to G$. Therefore, if $a,b\in G$ are such that $a=b$, then $(a,g)=(b,g)$ for every $g\in G$, and hence $\star(a,g)=\star(b,g)$, or (in infix notation) $a\star g=b\star g$. So, in groups, $a=b\Longrightarrow ag=bg$ for every $g$.