TITLE: In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?
QUESTION [9 upvotes]: In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column? 
It's obviously not the Closure axiom, and after some inspection, I believe the Inverses axiom does fail.  
However, I'm not so sure how to show whether or not the two other axioms fail (Identity and Associativity).

REPLY [12 votes]: 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:

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