Dummit-foote p.21
Let $G=\{g_1,\cdots,g_n\}$ be a finite group with $g_1=1$.
The Cayley table of $G$ is the $n\times n$ matrix whose (i,j)-entry is $g_ig_j$.
This definition is based on an order of $G$.
If one gives another order on $G$, then the Cayley table of $G$ would not be identical to the original.
Q: Is there any way to make the concept of Cayley table precise?
This table is useful to visualize small(informally speaking) finite groups.
However, using this table to study them is just the same thing as using geometry in topology or analysis.
What are your thoughts?