In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?

None of the properties fail automatically

For any group property, you can always find a Cayley table where there's a duplicate entry and yet that property still holds. Here are examples for each:

1. Associativity can still hold. $$\begin{array}{c|cc} \times& 0 & 1 \\\hline 0 & 0 & 0\\1 & 0 & 1\end{array}$$

2. Identity can still hold. (Same example.) $$\begin{array}{c|cc} \times& 0 & 1 \\\hline 0 & 0 & 0\\1 & 0 & 1\end{array}$$

3. Inverses can still hold. (Here, $a$ and $b$ are inverses of each other.)$$\begin{array}{c|ccc} & e & a & b \\\hline e & e & a & b\\a & a & a & e \\b & b & e & a\end{array}$$

However, either associativity fails or inverses fail.

If there's a duplicate row, then $ab=ac$ for some $b\neq c$. Suppose the operator has inverses and associativity. Then we get $a^{-1}ab = a^{-1}ac$ so that $b=c$— contradicting our assumption that $b\neq c$.

So if there's a duplicate row, the operator can either be associative (as shown above), or have inverses (as shown above), but never both.

For confirmation, note that in the example tables above, #1 is associative but not invertible because of 0, and #3 is invertible but not associative because $(bb)a = aa = a$ but $b(ba) = be = b$.)

Diagram

Groups can't have repeat entries. Therefore, if a table has repeat entries, it's not a group. If it's not a group, then it's not in the green region of this diagram. Visually, you can see that such a table can't be both associative and have inverses at the same time. And you can show that there exist tables with duplicate rows that belong to any other non-green region of this diagram.

If you have a monoid, an entry can appear twice in a row or a column. However, the axioms "Identity" and "Associativity" are satisfied. Suppose now that $xa = xb$ with $a \not= b$. Then $x$ cannot have an inverse, since otherwise $xa = xb$ implies $x^{-1}xa = x^{-1}xb$, that is, $a = b$.