Suppose $G_1,G_2$ are two finite groups and their character tables are the same. Can I prove or disprove that $|G_1|=|G_2|$?
I have just started on character theory and I have been thinking proving this using row orthogonality and maybe also column orthogonality.
I have developed some thoughts using row orthogonality but I have not really gone far. Suppose $\{g_1,\cdots g_n\}$ is the complete set of conjugacy classes of $G_1$ and $\{h_1,\cdots,h_n\}$ is the conjugacy class of $G_2$. WLOG assume $g_i$ and $h_i$ are arranged such that they are in increasing order of cardinality so $|g_1|\leq|g_2|$ etc. Then my instinct is telling me that to satisfy the row orthogonality we needed $|h_i|=k|g_i|$ for some $k\in\mathbb{N}.$ However I am not sure if this is true or if there is a way of proving so.
How could I proceed? Any hint is much appreciated!
Hint: use the formula: $\sum_{\chi \in Irr(G)}\chi(1)^2=|G|$ (which basically is an orthogonality relation)