Answer to Question #162933 in Abstract Algebra for K

Question

Give a proof that the associative property is a structural property. (Start with two isomorphic structures, let the operation on one be associative and show that its image must also be associative).

Answer 1

Let $(G,*)$ and $(K,\circ)$ be two isomorphic structures, and the operation $*$ on $G$ is associative. Let us show that the operation $\circ$ on $K$ is also associative. Let $\psi: G\to K$ be an isomorphism, and $a',b',c'\in K$ be arbitrary. Since $\psi$ is surjection, there exists $a,b,c\in G$ such that $a'=\psi(a), b'=\psi(b), c'=\psi(c).$ It follows that

$a'\circ[b'\circ c']=\psi(a)\circ[\psi(b)\circ \psi(c)]=\psi(a)\circ\psi(b*c)=\psi(a*(b*c))= \psi((a*b)*c)=\psi(a*b)\circ\psi(c)=[\psi(a)\circ\psi(b)]\circ\psi(c)=[a'\circ b]'\circ c'$

Therefore, the operation $\circ$ on $K$ is also associative, and  the associative property is a structural property.