Cayley tables for two non-isomorphic groups of order 4. Not sure how to make tables, but:
For a binary operation $*$, and set $\{e,a,b,c\}$, in the Cayley table, $a*a$ can be filled with either the identity or an element different from both $e$ and $a$. If in the table the place for $a*a$ is filled with $e$, then the rest of the table can be filled out in two different ways.
What are the two different ways?
Thanks
 A: If $a * a = e$, then the following are also determined (since each row/column of the Cayley table contains each group element exactly once): $$b * a = a * b = c; \quad c * a = a * c = b.$$ It follows that the only remaining possibilities for $b*b$ are $e$ and $c$, and we can extend each of these (in exactly one way) to give the Cayley table for a group.
A: Two days ago I computed all Cayley tables for the Gorup $(G,*)$ with $|G|=4$ and came up with 4 distinct Cayley tables (!):
For the abelian group of the itegers modulo $4$: $(\mathbb{Z}/(4),+_4)$.
$1.$ $$\begin{array}{|c|c|c|c|c|}
\hline
+_4 & e & a & b & c\\
\hline
e & e & a & b & c \\
\hline
a & a & b & c & e \\
\hline
b & b & c & e & a \\
\hline
c & c & e & a & b \\
\hline
\end{array}$$
For the Group formed by the $\mathbb{C}$-roots of $z^4=1$: $(\xi_4,\cdot_\mathbb{C})$
$2.$ with $\xi_4=\{ 1,i,-i,-1 \}$
$$\begin{array}{|c|c|c|c|c|}
\hline
\cdot_\mathbb{C} & e & a & b & c\\
\hline
e & e & a & b & c \\
\hline
a & a & c & e & b \\
\hline
b & b & e & c & a \\
\hline
c & c & b & a & e \\
\hline
\end{array}$$ 
$3.$ with $\xi_4=\{1,-1,i,-i \}$
$$\begin{array}{|c|c|c|c|c|}
\hline
\cdot_\mathbb{C} & e & a & b & c\\
\hline
e & e & a & b & c \\
\hline
a & a & e & c & b \\
\hline
b & b & c & a & e \\
\hline
c & c & b & e & a \\
\hline
\end{array}$$
And for the Klein-$4$-Group: $(\mathbb{Z}_2\times\mathbb{Z}_2,+_{\mathbb{Z}_2\times\mathbb{Z}_2})$
$4.$ $$\begin{array}{|c|c|c|c|c|}
\hline
+_{\mathbb{Z}_2\times\mathbb{Z}_2} & e & a & b & c\\
\hline
e & e & a & b & c \\
\hline
a & a & e & c & b \\
\hline
b & b & c & e & a \\
\hline
c & c & b & a & e \\
\hline
\end{array}$$
The last two are the ones that you are looking for.
