group isomorphisms not decomposable into relabelling or row-col exchange sequences

Question

I am interested in the way groups represent abstract structure, but unfortunately I'm dyslexic in understanding the notation. As a novice in this area I see statements that group isomorphisms "commonly" result from relabelling the elements, or performing row/column swaps on the Cayley table. However, it is not clearly stated whether these actions can generate all isomorphisms. So my question is: can a finite group isomorphism ALWAYS be decomposed into a relabelling followed by a sequence of row/column exchanges?

Answer 1

$\renewcommand{\phi}{\varphi}$It might not be a particularly useful point of view but, yes, this is indeed the case.

Let $G$ be a group, and $g, h \in G$. In the $g$-th row, $h$-th column of the Cayley table you will find $g h$.

Now let $\phi$ be an automorphism of $G$. Apply $\phi$ to the labels and the entries of the Cayley table. Now in the $\phi(g)$-th row, $\phi(h)$-th column you have $\phi(g h)$, which equals $\phi(g) \phi(h)$, as $\phi$ is a homomorphism.

So apart from a permutation (relabelling) of rows and columns, you got the original Cayley table back.

If instead of an automorphism $\phi$ of the group $G$ you have an isomorphism $\phi : G \to H$ from $G$ to another group $H$, the new table is indeed the multiplication table of $H$.