Simple question about finite groups

Question

If two finite groups are isomorphic, does that mean they have the same character table?

Answer 1

The answer is yes.

A character $\chi$ corresponding to a representation $\rho:G\to \mathrm{GL}(V)$ is the trace of the representation. That is $\chi(g)=\mathrm{Tr}(\rho(g))$. If $G\cong H$ via $\alpha:G\to H$ then $\chi\circ\alpha^{-1}$ is a character of $H$ corresponding to the representation $\rho\circ\alpha^{-1}$. Moreover $\chi\circ\alpha^{-1}(\alpha(g))=\chi(g)$, so the entries in the character table corresponding to $\chi\circ\alpha$ will be the same as those corresponding to $\chi$. Using this correspondence we see that (up to permuting rows and columns of the table) the character tables of $G$ and $H$ are the same.