Character tables of isomorphic groups

Question

Let us consider two groups $G$ and $G'$ which are isomorphic to each other. Since isomorphic groups can be considered as same upto isomorphism, is the character table same for both groups?

Answer 1

As Derek Holt said in the comments, both tables are equal up to rows and columns permutations. To prove the assert, consider an isomorphism $\sigma:G'\to G$ and an irreducible character $\chi\in\text{Irr}(G)$. If $\mathcal X$ is a representation of $G$ affording $\chi$ then $\mathcal X\circ\sigma$ a representation of $G'$ (since it is a composition of homomorphisms) affording $\chi\circ\sigma$. Now, since $\sigma$ is a bijection it holds $$[\chi,\chi]=\frac{1}{|G|}\sum_{g\in G}|\chi(g)|^2=\frac{1}{|G'|}\sum_{g'\in G'}|\chi(\sigma(g'))|^2=[\chi\circ\sigma,\chi\circ\sigma].$$ Hence, $\overline{\sigma}:\text{Irr}(G)\to\text{Irr}(G')$ defined as $\overline{\sigma}(\chi)=\chi\circ\sigma$ is well-defined, and it is a bijection since its inverse is the function $\psi:\text{Irr}(G')\to\text{Irr}(G)$ such that $\psi(\chi')=\chi'\circ\sigma^{-1}$ for each $\chi'\in\text{Irr}(G')$. The result follows from the fact that group isomorphisms preserve conjugacy classes.