How did the physicists in the 1920s become aware of the importance of group theory in quantum mechanics? Was group theory already part of the physics curriculum at that time, perhaps in connection to crystallography?
From Pais's book Inward Bound, chapter 13:
Wigner had become interested in $n>2$ identical particle problem. He rapidly mastered the case $n=3$ (without spin). His methods were rather laborious; for example, he had to solve a (reducible) equation of degree six. It would be pretty awful to go on this way to higher $n$. So, Wigner told me, he went to consult his friend the mathematician Johnny von Neumann. Johnny thought a few moments then told him that he should read certain papers by Frobenius and by Schur which he promised to bring the next day. As a result Wigner's paper on the case of general $n$ (no spin), was ready soon and was submitted in November 1926. It contains an acknowledgement to von Neumann, and also the following phrase: "There exists a well-developed mathematical theory which one can use here: the theory of transformation groups which are isomorphic with the symmetric group (the group of permutations)".
Thus did group theory enter quantum mechanics.
The introduction of group theory was not universally welcomed. In the preface to Sternberg's Group Theory and Physics, he quotes from the autobiography of John Slater, head of the MIT physics department for many years (p.60-62):
It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their "Gruppenpest": the pest of the group theory... The authors of the "Gruppenpest" wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practicle consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest.
I had what I can only describe as a feeling of outrage at the turn which the subject had taken...
As soon as this [Slater's] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as "Slater has slain the 'Gruppenpest'". I believe that no other piece of work I have done was so universally popular.
Sternberg comments that this was not an atypical reaction, but adds "It is, however, amazing to consider that this autobiography was published in 1975, after the major triumphs of group theory in elementary particle physics."
Wigner writes in the preface to the 1959 English edition of his Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra:
When the original German version was first published, in 1931, there was a great reluctance among physicists toward accepting group theoretical arguments and the group theoretical point of view. It pleases the author that this reluctance has virtually vanished and that, in fact, the younger generation does not understand the causes and basis for this reluctance. Of the older generation it was probably M. von Laue who first recognized the significance of group theory as the natural tool with which to obtain a first orientation in problems of quantum mechanics.
...The initial stimulus for these articles [on which this book is based] was given by the investigations of Heisenberg and Dirac on the quantum theory of assemblies of identical particles.