Cayley table property of an infinite group A Cayley table of an finite group has to have every element exactly once in every row and exactly once in every column.
Proof that every element of a group has to be at most once in every row and at most once in every column:
Let $(G, \circ)$ be a group and $a, b, c, d \in G$ with:
(I) $a \circ b = d$
(II) $a \circ c = d \Leftrightarrow a = d \circ c^{-1}$
Then:
$\begin{align}
(a \circ c) \circ (a \circ b)    &= d \circ d \\
\Leftrightarrow d \circ (d \circ c^{-1} \circ b) &= d \circ d \\
\Leftrightarrow d \circ c^{-1} \circ b           &= d\\
\Leftrightarrow c^{-1} \circ b &= e\\
\Leftrightarrow b &= c
\end{align}$
As the group is finite, this also means it is exactly once in every row/column ($\forall a,b \in G: a \circ b = x$ with $x \in G$).
Now my question is:
Does a group with an infinite number of elements exist, that has not every element in every row/column of its Cayley table?
(I know that Cayley tables usually get used only for finite groups. But if set of the group has a countable number of elements, you can imagine a Cayley table. For example, $(\mathbb{Z}, +)$ has obviously every element in every row/column).
 A: The term Cayley table is generally restricted to finite groups. However, it’s certainly possible to generalize the idea. For a group $G$ and an element $a\in G$, the $a$ ‘row’ of the table is essentially just the function $$f_a:G\to G:b\mapsto ab\;,$$ and the $a$ ‘column’ is essentially just the function $$f^a:G\to G:b\mapsto ba\;.$$ If $G$ is countably infinite, you can visualize the Cayley table as an infinite matrix.
Let $G$ be any group, and fix $a\in G$. For each $b\in G$ you have $b=a(a^{-1}b)$, so $b$ appears in row $a$ in column $a^{-1}b$. Similarly, $b=(ba^{-1})a$, so $b$ appears in column $a$ in row $ba^{-1}$. It follows that $b$ appears in every row and column. The cardinality of the group doesn’t matter.
Added: You didn’t ask, but it’s also clear that each element of $G$ appears only once in each row and column: if $ax=ay$ or $xa=ya$, then $x=y$. Thus, each of the maps $f_a$ and $f^a$ for $a\in G$ is a bijection from $G$ onto itself, i.e., a permutation of $G$. The set of all permutations of $G$ is denoted by $\operatorname{Sym}(G)$ and is a group under composition of functions; the maps
$$G\to\operatorname{Sym}(G):a\mapsto f_a$$
and
$$G\to\operatorname{Sym}(G):a\mapsto f^a$$
are isomorphisms of $G$ to subgroups of $\operatorname{Sym}(G)$. This is Cayley’s theorem.
A: In the general setting, having at least one element in every row and column just means that given any $g$ and any $a$, there exists a $b$ and $b'$ such that $ab = g$ and $b'a = g$. This is true because you can let $b = a^{-1}g$ and $b' = ga^{-1}$. 
Having exactly one element in each row or column is equivalent to $ac = g$ and $ac' = g$ implying that $c = c'$. This is because $ac = g  = ac'$. Multiplying $a^{-1}$ to both side gives $c = c'$. Do the same thing for $ca = g$ and $c'a = g$. 
