The authors give a negative answer to the long-standing question whether or not the set of all elations about some point \(x\) forms a group for any thick generalized quadrangle \(S^{(x)}\) having \(x\) as an elation point or a center of transitivity. Furthermore, an answer is given for each of the known generalized quadrangles. In particular, for any nonclassical translation generalized quadrangle \(\mathcal S\) of order \((s,t)\) with \(s,t > 1\) and any elation point \(x \in {\mathcal S}^D\) they prove the following results for the set \(G_{x}\) of elations about \(x\): (1) If \(s=t\) is even, then \(G_{x}\) is a group, (2) If \(t = s^2\) is even and \({\mathcal S}\) is a \(T_{3}({\mathcal O})\) for some ovoid \(\mathcal O\) of PG\((3,s)\), then \(G_{x}\) is a group. (3) If \(t=s^2\) is odd and \( \mathcal S\) is a good translation generalized quadrangle, then \(G_ {x}\) fails to be a group.